Network flow, transportation, and scheduling; theory and algorithms

Network Blow, Transportation and Scheduling Theory and Algorithms

Network Flow,Transportation and Scheduling Theory and Algorithms MASAO IRI DEPARTMENT OF MATHEMATICAL ENGINEERING AND INSTRUMENTATION PHYSICS FACULTY OF ENGINEERING UNIVERSITY OF TOKYO TOKYO, JAPAN

@

ACADEMIC PRESS

New York and London

1969

COPYRIGHT 0 1969, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED NO PART O F THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT W R I T E N PERMISSION FROM THE PUBLISHERS.

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United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W1 X 6BA

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 69-13483 AMS 1968 SUBJECT CLASSIFICATIONS 0540,9430

PRINTED IN THE UNITED STATES OF AMERICA

From the theoretical points of view, network-flow problems constitute a very special subclass among general mathematical-programmingproblems -special in the sense that the constraint relations for variables are determined by the topological properties of a geometrical figure called a graph and that the objective functions are separable in form. From the points of view of practical applications, they not only serve as fairly good models of various kinds of problems, such as transportation processes, personnel assignment, project scheduling, etc., but also as powerful means to deal with some kinds of graph-theoretic as well as combinatorial-mathematical problems. Furthermore we can take advantage of their special characters to solve within practicable time and cost such large-scale network-flow problems that the solution of a general mathematical-programming problem of similar scale might prohibitively be difficult. The present book is intended to give a self-contained exposition of the theory and to develop thereupon the practical algorithms for networkflow problems. Therefore, it is basically of expository character although there will be found a lot of new material. Applied mathematicians, postgraduate students, and theoretical-minded people engaged in operations research, system engineering, electrical and communication engineering, etc. will be the audience. There are many alternative ways in which to expound the theory of network flows. Emphasis is sometimes laid upon the mathematicalprogramming facet of the problems and sometimes upon the graphtheoretic. It would not be impossible to give an ad hoc exposition even without sound preliminaries. However, for the deep understanding of the V

vi

Preface

whole theory and for the perfect command of the techniques of dealing with the problems, it seems necessary to prepare preliminary explanations on convex sets and convex functions together with considerable details about the topological properties of linear graphs, as we shall do in Chapters 11 and 111. Two seemingly contradictory ways of exposition-the general theoretical treatment and the practical algorithmic treatment of the problem-are in fact mutually complementary ways of approaching the true understanding of the problems. We have two chapters, i.e. Chapters IV and V, one of which is devoted exclusively to the theoretical treatment of general network-flow problems while the other emphasizes the algorithmic treatment of linear network-flow problems. The appendices contain a survey of topics and research problems in the related fields as well as mathematical supplements and suggestions on digital and analogue computations. The prerequisites to this book are rudiments of linear algebra, differential and integral calculus, set theory and elementary topology. Although there is an abundance of electrical terminology in this book, no knowledge of electricity or of electric networks will be assumed as the background. (Of course, physical intuition will help understand the problem.) The mathematical notations adopted in this book are more old-fashioned than modern, because the correspondence between mathematical concepts and their physical counterparts may become intuitively more intelligible in that manner. As for notations, particular attention should be paid to the relative positions of indices to kernel letters since they are of some significance in almost all cases. In the existing literature we have two excellent and pioneering books on network-flow problems, i.e. one by L. R. Ford, Jr., and D. R. Fulkerson and the other by C. Berge and A. Ghouila-Houri (see Ford and Fulkerson [I] and Berge and Ghouila-Houri [l] in the reference list at the end of this book), which have made a great contribution to the progress of science and to which the present book also owes not a little. The present book is based primarily upon a series of the author’s own investigations-among which we should mention, besides those found in the reference list, the following articles : “ Network Theory Based upon Topology.’’ RAAG Research Notes, Second Series, No. 10 (1956). “An Application of the Theory of Boolean Lattices to the Problems of Oriented Graphs or On-Off Circuits Containing Rectifiers.” ibid., No. 14 (1956). “ O n the Torsion and Linkage Characteristics and the Duality of

Preface

vii

Electric, Magnetic and Dielectric Networks (with Y. Mizoo and K. Kondo).” RAAG Memoirs, Vol. 2 (1958), A-VIII, pp. 84-117. “ Theory of Communication and Transportation Networks (with T. Sunaga).” ibid., G-11, pp. 444-468. “Algebraic and Topological Foundations of the Analysis and Synthesis of Oriented Switching Circuits.” ibid., G-111, pp. 469-518. “Algebraic and Topological Theory of the Problems of Transportation Networks with the Help of Electric Circuit Models.” RAAG Research Notes, Third Series, No. 13 (1959). “A New Method of Solving Transportation-Network Problems.” Journal of the Operations Research Society of Japan, Vol. 3, No. 112 (1960), pp. 27-87. “ Fundamentals of the Algebraical and Topological Treatments of General Information Networks.” RAAG Memoirs, Vol. 3 (1962) G-V, pp. 418-450. “ Several Applications of the Basic Theory of General Information Networks-Connexion Properties of Graphs and Finite Deterministic Two-Person Games.” ibid., G-VI, pp. 451-471. “ Topological and Algebraical Theory and Methods of Transportation Networks.” ibid., G-VII, pp. 472-494. “ On the Number of 2-Isomorphic Graphs.” RAAG Research Notes, Third Series, No. 72 (1963). “A Criterion for the Reducibility of a Linear Programming Problem to a Linear Network-Flow Problem.” ibid., No. 98 (1966). “ Metatheoretical Considerations on Duality.” ibid., No. 124 (1968). “ On the Topological Conditions for the Realization of Multi-Port Networks.” RAAG Memoirs, Vol. 4 (1968), A-XV, pp. 47-61. Some of the results published in the works of J. B. Dennis and of G. J. Minty (see Dennis [l] and Minty [l]) are included and Zoutendijk’s book [11was useful. However, the recent works of R. T. Rockafeller whose standpoint seems to have much to do with ours could not be taken into account because of time limitations (cf. Rockafeller [13). It is a matter of regret that the geographical and other constraints under which the author lies made it difficult to compile a complete literature list, to a fatal drawback of this book. It is a pleasure to record the author’s gratitude to those to whom he is indebted, directly or indirectly, in writing this book. First of all, my thanks are due to Professor Richard Bellman who kindly invited and encouraged me to write a book in the “ Mathematics in Science and Engineering” series. I would like to take this opportunity also to express

...

Vlll

Preface

my deep sense of gratitude to Professor Kazuo Kondo of the University of Tokyo under whose guidance 1have studied and worked since my student days, to the members of the RAAG (Research Association of Applied Geometry, Japan) and other friends at home and abroad whose stimulative discussions and suggestions have been very helpful, and to Professor Sigeiti Moriguti of the University of Tokyo who guides me in Operations Research, Computer Applications and Numerical Methods. It should be recorded here also that, in December 1965, the Matsunaga Science Foundation awarded me the Matsunaga Prize for the “Researches on Network Problems in Operations Research,” which greatly stimulated me in writing this book. Finally, I must fulfill my private duty to mention the invaluable collaboration of my wife, Mrs. Yumi Iri, who did not only type all the manuscript but also suggested many formal and substantial improvements in it. IRI MASAO June I969

Preface

V

I. Introduction 1. Transportation Problems 2. Project Scheduling Problems

3. Electric Networks with Nonlinear Resistors

1 3 5

11. Convex Sets and Functions 4. Convex Sets

5. Convex Functions 6. Minimum of a Convex Function on a Convex Set

8 11

26

111. Topological Properties of Networks 7. Graphs and Vector Spaces 8. Connectivity 9. Loops, Cutsets, Trees, and Cotrees 10. Duality 11. 2-Isomorphism 12. Planar Graph and Its Dual

28 37

41 66

67 73

IV. Network-Flow Problems in General 13. Constituents of a Network 14. Remarks on Duality

75 87

ix

Contents

X

15. 16. 17. 18. 19.

Minimization Problems on a Network General Properties of Solutions Two-Terminal Characteristic Problems of Approximation Equivalence Transformations of Networks

87 101 110 116 120

V. Linear Network-Flow Problems 20. Maximum Flow 21. Minimum Route 22. Re!ations between the Maximum-Flow Problem and the Minimum-

Route Problem 23. Two-Terminal Characteristics 24. General Linear Network-Flow Problems

130 174 192 193 243

Appendices 1 : Farkas’ Theorem 2: Equisignum Decomposition of a Vector 3: Theory of Linear Resistor Networks 4: Digital and Analog Computations 5: Reducibility of a Linear Programming Problem to a Linear Network-Flow Problem 6: Related Problems and Extensions of Network-Flow Problems

256 263 267 274

References

307

Index

311

288 296

Introduction

A network-flow problem is, mathematically, a special case of mathematical programming problems, in which the constraint relations imposed on variables are intimately connected with a graph, i.e., with a geometrical figure consisting of points and lines. However, we believe it in order to give a rough sketch of the essence of the problem from the point of view of practical applications and then to enter into its mathematical treatments. Thus, we shall begin with rough formulations of the three typical physical problems of which the network-flow problem is a mathematical model. 1. TRANSPORTATION PROBLEMS

Let us consider transporting a commodity through a network of routes such as shown by solid-line arrows in Fig. 1.1. We assume that there is no stagnation of commodity flow at nodes, i.e., at junction points of routes; we denote by r" the rate of commodity flow through the Kth route, and assume that the total cost z for a transportation plan is equal to the sum of the costs related to the routes, each of which is a given function rp,(r) of p,i.e.,

1

2

I. Introduction

Our problem is to obtain a transportation plan, i.e., a set of values .. ., t5),according to which a given rate c of commodity flow is made from the entrance node 1 to the exit node 4 through the network, and which, among all such plans, minimizes the total cost z. ({I,

Fig. 1.1. Transportation network.

Instead of dealing with the problem as it was just posed, we make a modification as follows. We introduce a new fictitious route 6 (drawn by a dashed-line arrow in Fig. 1.1) which connects the exit directly to the entrance and through which the flow going out of the network at the exit returns to the entrance again to enter the network. The condition that the total rate of commodity flow from the entrance to the exit be equal to c can now be written as 56

= c,

where t6 is the flow rate in route 6 . Let (P6(t6) be a function whose value is 0 when t6 = c and is very large when t6 # c. Then our problem would be equivalent, at least approximately, to the problem of obtaining (<', .. .,C6) which minimizes 6

=

c

(PK(<">

K = l

(14

without the condition t6 = c. Thus, our problem may be formulated mathematically as that of obtaining the values of 5K's which, subject to the nonstagnation condition

c DKOltK=O 6

K=

1

(a=l,...,4),

3

2. Project Scheduling Problems

minimizes I [in (1.2)], where

cm=

1 1 1 2 1-1

2 1 0

0 4 3 1 0

-1 0

4 0 1 -1 0 0 - 1 - 1 3 0 1

5 6 0 - 1 0 0

(1.4)

0 1

1

is the matrix whose element 6; is equal to 1 if route K starts from node a, to - 1 if route K ends at node a, and to 0 otherwise. ... , C4 corresponding to the Introducing the Lagrange multipliers four equalities in (1.3) (the applicability of the method of the Lagrange multipliers being assumed) and defining

cl,

6

6

4

~ ( t "Ca), = C1 (PAY)- aC =l

(a

K=

C 6/tK, ~ = l

(1.5)

we can determine the solution of our problem by putting the partial derivatives of F equal to 0, i.e., we have 6

4

aF/atK= (~~'(5") - C 6/ca= 0, a= 1

aF/ay, =

- C 6:tK= 0, K=l

or, putting

we have 6

I

I

I

(a = 1,.

..,4),

(K = 1,.

. . ,6),

A

qK= ?K

C D/C,

a=l

= q,'(cK)

( K = 1,

..., 6).

2. PROJECT SCHEDULING PROBLEMS

Figure 2.1, which is geometrically the same as Fig. 1.1 in essence, is an example of the so-called " arrow diagram," i.e., a diagram representing the precedence relation of jobs constituting an entire project. It consists of five solid-line arrows, each corresponding to a job, where a job whose

4

I. Introduction

correspondingarrow starts from a node cannot begin until all the jobs whose corresponding arrows end at that node have finished. Furthermore, the entire project starts at node 1 and is completed at node 4. We make correspond to each node a a time (instant) C, by which all the jobs (i.e., the corresponding arrows) ending at node a have been completed and after which the jobs starting therefrom begin. If job K starts from node a

Fig. 2.1. Arrow diagram of a project.

and ends at node p , we assign to it a time (duration) - q K = Cfi - C, ,calling -qK’s job times. A job time thus defined includes, in general, inefficient time beside the efficient time during which the job is actually being carried for job K to be completed out. It is assumed that it costs as much as v(q,) in time -q. and that the total project cost is

Our problem is to obtain a time schedule of the project, i.e., an assignment of job times to the constituent jobs, such that the time C4 - Cl required by the entire project is equal to a given value e and that the w is as small as possible. In the manner of $1, we shall introduce a new fictitious job or arrow connecting node 4 to node 1 (shown by a dashed line in Fig. 2.1) as well as its time -q6 = Cl - c4. If i,b6(q6) is a function that is equal to 0 when = e but is very large otherwise, then the minimization of w under the condition c4 - Cl(= q 6 ) = e will be equivalent to the minimization of

3. Electric Networks with Nonlinear Resistors

5

without that condition. Thus, our problem may mathematically be formulated as that of obtaining the values of q,'s and Cis which, subject to the continuity relation of time 4

qK =

C D/Ca a= 1

(K =

1, ... , 6 ) ,

minimizes 5 [in (2.2)], where D/ is the same as that defined in $1 [see (1.411. Introducing the Lagrange multipliers tl, . . . , corresponding to the six equalities (2.3) and defining

r6

we can solve the problem by the equations

or 6

(a= 1,. .. ,4),

Apparently, (2.5) is quite the same in "form" as (1.7). The only difference is that in (1.7), q, is related to 5" by a function of t", whereas in (2.5), 5" is expressed as a function of q K . 3. ELECTRIC NETWORKS WITH NONLINEAR RESISTORS

Let us consider an electric network with six branches and four nodes whose connection relations are as shown in Fig. 3.1. The current in branch K will be denoted by t", the voltage across it by q, , and the potential at node a by Ca (a= 1, .. ., 4 ; K = 1, .. ., 6 ) . In terms of the same matrix

6

I. Introduction

+ h i 4

Fig. 3.1. Nonlinear resistor network.

b,oI that

we made use of in $1 and $2, Kirchhoff's current law in electric network theory is expressed as 6

1 6,"y=o K=

1

(Ct=l,..., 4),

(3.1)

which states that there is no stagnation of current at any node. Kirchhoff's voltage law is written as 4

v], =

C oC,"ca

(K =

1,. ..,6),

a= 1

which states that the voltage across a branch is equal to the difference of 1

(Voltage)

Fig. 3.2. Current-voltage relation.

3. Electric Networks with Nonlinear Resistors

7

the potentials at the two endnodes of that branch. In addition, we assume that the current (" in a branch and the voltage q K across it are related to each other by a function (see Fig. 3.2)

(IC= 1, . . . ,6). Here arises a question of how to obtain a set of values (g',

5" = g"(qK)

qK = rK(tK)

(3.3) .. . , t 6 ; ql, . ..,q 6 ; Cl, .., c4) which satisfy (3.1), (3.2), and (3.3) simultaneously. Thus, the problem of an electric network with nonlinear resistors will be of entirely the same kind as that of a transportation network (1.7) and of a project scheduling (2.5).

.

or

AA

Convex Sets and Functions

In discussions of minimization problems, we often encounter the concepts of " convex sets " and " convex functions." It is true that, in not a few cases, the cost function to be minimized is convex in regard to the controllable variables, but it cannot be denied that we confine ourselves to such sets and functions because, by so doing, the theoretical treatments are made easier and more fruitful. This chapter will be devoted to a brief summary of fundamental concepts and theorems regarding convex sets and functions. 4. CONVEX SETS

By the n-dimensional Euclidean space R" we mean a vector space, or module, V(R, {el, . . . ,en})with el, . . . ,enas the basis vectors and the field of real numbers R as the coefficient domain (and of which the topology is the nth power of the natural topology of R). Definition 4.1. A subset 6 of the n-dimensional Euclidean space Rnis said to be convex if, for arbitrary two points

8

9

4. Convex Sets

in it, every point on the straight line segment connecting them belongs to it, i.e., if

5 = (1 - 1)S + 15 1

0

n

=

C

1

[(I - 1)t“- 1t“]e, 1

0

K=l

belongs to (I: for all 1 (0 6 1 6 1).

R”itself, as well as the empty set, is convex. Definition 4.2. Let

5,. . ., 6 be 1

m points in R” and o,. .., o be m 1

m

m

nonnegative real numbers whose sum is equal to 1 : m

pD=l.

w20, i

By the weighted mean of

i=l

i

5’s with weights o ’ s we mean the point i

i

c 05. m

5=

Theorem 4.1. 6 (g R”)is convex if and only if for arbitrary points E (I: and weights w, . . .,o ,the weighted mean

5, . ..,& 1

(4-3)

i=li i

1

m

m m

5=

105

i=l i i

also belongs to 6. PROOF. “If” is obvious since we have the very definition of a convex set by putting rn = 2, o = 1, and o = 1 - 1. To prove “only if,” we 1

2

may assume, without loss in generality, that every o is positive, because i

we may omit the terms with the null weight from (4.3). Put

Then we have

10

11. Convex Sets and Functions

where (O
Therefore, if

c

E (5,

then

I- 1

5 E (5. Since 5 = 6 E (5, 5 = 6 E (5. I

1

Q.E.D.

m

1

In Fig. 4.1, examples of convex and nonconvex sets are shown.

(b)

(a)

Fig. 4.1. (a) Convex set, and (b) nonconvex set.

The following theorem will be obvious from the definition. Theorem 4.2. If (5,, (5, , . . . are convex (their number may be finite or infinite), then so is their intersection El n (5, n * * * .

It should be noted that the union (5, u (5, of convex sets (5,, (5, is not always convex. For a nonconvex set (5, its convex hull is defined to be the smallest convex set containing 6,which is uniquely determined by virtue of Theorem 4.2. Theorem 4.3. The intersection of a finite or infinite number of closed convex sets is a closed convex set. The intersection of a finite number of open convex sets is an open convex set. PROOF. Recall that the intersection of a finite or infinite number of closed sets is closed and that the intersection of a finite number of open sets is open. Q.E.D.

If the set of indices ( 1 , . . . , rn and ( K , + ~ , . . . , K,+,),

+ n) is divided into two sets ( K ~ .,. . , K,) m+n

by the projection of a point

the ( t i l , . , . , tirn) subspace, we mean the point

6 = C tKeKE Rm+"to

...

K= 1

m

=

1 FieKi R", and by the E

i= 1

11

5. Convex Functions

projection of a subset 6 ( E R"'") to the same subspace, we mean the subset ( E R") of points 5 such that there exists a point in 6 of which the projection is 5. The following two theorems are given without proof since they are easy to prove on the basis of the definition of convex sets and of the elements of the theory of Euclidean spaces.

Theorem 4.4. The projection of a convex set in R"'" to a (K,, . . . , K,) subspace R" is also convex in R". Theorem 4.5. The direct product of convex sets is convex. A convex set is obviously connected, so that we have the following nine kinds of nonempty convex sets in the one-dimensional case, where b and c are finite real numbers. They are called intervals in R' = R.

Cb, cl = (51b 5 5 s c > (b s 4,

(-a, ~o)=R,

cl = (515 2 c>,

Cb, 00)

= {elb

2 51,

(b, C I

= {51b

< 5 c> (b < 4,

(-00,

e

Cb, c) = (516 5 5 < c> (b < 4, (b, a)=

5 I b < t-1,

(-

= (515

< 4,

(b, c ) = {rib < 5 < c } (b < c).

The first four kinds are closed intervals, whereas the last three kinds and the first one are open intervals. 5. CONVEX FUNCTIONS

A nonmathematician will probably regard the graph of y = x2 as being " concave " and that of y = -x2 as being " convex." Curiously enough, the convention of mathematics requires one to do the converse, i.e., one should say that y = x 2 is a convex function, whereas y = - x2 is concave.

5.1. Convex Functions in General Let us begin with the definition of a convex function whose domain is in the n-dimensional Euclidean space.

Definition 5.1. A function f(4)=f(t', .. ., 5") defined on an open convex set 6 in the n-dimensional Euclidean space R" is said to be convex

12

11. Convex Sets and Functions

if, for arbitrary two points E, and E, in (1 - W

(5.

and an arbitrary 1(0 2 1

1

0

( 50 ) + W 51 )

If, for any mutually distinct points

-4 5 0

(5.1)

6, E, E 6,and any 1not equal to 0 or 1, 0

1

(1 - ME,)+ U ( 5 ) >fC(1 - 4 1

0

+ 251. 1

I),

5 + 151,

0

(5.1‘)

1

, f ( E , ) is said to be strictly convex. If --f(&) is convex or strictly convex,f(S) is said to be concave or strictly concave, respectively. The foregoing definition of convex functions might seem to be applicable to the functions whose domains are not necessarily open but only convex. However, we shall adopt a slightly more restricted definition of them in order to simplify the discussions about the behavior of the functions on the boundary of 6,especially about the continuity and the differentiability. Definition 5.2. A function f (6) defined on a convex set (5. (not necessarily open) is said to be convex if there exist an open convex set T, ( 2(5.) and a convex function g(6) defined on 3 such that

6 E (5..

for all

f(E,)= g(5)

(5.2)

The following two theorems are direct consequences of the definition and will require no formal proof. Theorem 5.1. A function defined on an open convex set (5. is convex if and only if for arbitrary points 6 , . . ., E, E (5. and arbitrary weights 1

m

m

w ) . . . )w ( f + l 2 0 , C w = 1 ) , 1

m

i=l i

i

m

i=l

i

i

i=l

(5.3)

i i

Theorem 5.2. If the domains of (a finite or infinite number of) convex functions .f;(E,), f2(E,), . . . are El, E2,. . . , respectively, then, for arbitrary nonnegative CI’S, the functions i

f(S) = 1 mfi(E,) i i

and

g ( 5 ) = SUPL(S> i

(5.4)

Ci. Moreover, if at least

are (so long as they exist) convex functions on i

13

5. Convex Functions

one f i ( Q is strictly convex with the corresponding a strictly positive, then i

f(6) is strictly convex.

Next, we shall change to the discussions on the continuity and the differentiability of convex functions.

Theorem 5.3. A convex function continuous at every point in (5..

f(6) on a convex set

(5.

in R" is

PROOF. It suffices to consider the case where (5. is open in view of Definition 5.2. Let 6 be an arbitrary point in (5. and E be an arbitrary positive number. Since (5. is open, there exists a positive number p such that all the points 6 2 pe, ( K = 1, . . . , n) belong to 6,where eK is the Kth basis vector. Put = max[max

lf(6 + peK) -f(6>1?

K

lf(6 -

peK>

-f(6>11?

(5'5)

K

and 6 = p/(2n) min(1, E / w ) ,

(5.6)

n

and choose a point x

=

1 xKeKsuch that IxK- lK1< 6 for every K . Then, K=l

we have

where y" E xK- 5" (I yKI < 6) and sgn y = 0, or < 0, and, conversely,

[Note that x belongs to follows that

(5.

= 1,

0, or - 1 according as y > 0,

by virtue of (5.7).] From (5.7) and (5.3), it

or

(5.9)

14

11. Convex Sets and Functions

Similarly, from (5.8) and (5.3), it follows that

or (5.10)

Thus, we have shown that for any I.f(x) -f(k)I < E

( > O ) , there exists a 6 ( > O ) such that

E

for every x with

- tK1< 6.

Ix"

Q.E.D.

(5.11)

Before entering into the discussion of differentiability, we first show Lemma 5.2. Letf(5) be a convex function on a convex set 6 (gR"), an arbitrary vector in R", and a and p be real numbers such that a > p and that

5 E 0, r be

5 + ar € 6

6 + pr € 0 .

and

Then F(u,

=

CJ'(6 + 4 - A 6 + fir)l/(a - P)

(5.12)

is monotone nondecreasing as a function of a as well as a function of

PROOF. Let

E,

a, a be real numbers such that a > a

1

2

1

3

6 + ar E 6

(i

=

2

p.

> a and that 3

1,2,3).

i

Then we have a-a

1

a-u 1

3

3

(5.13) may be rearranged as

i.e.,

F(u, a ) 2 F(a, a) I

3

2

3

(a > a 1

2

> a), 3

(5.14)

15

5. Convex Functions

and, at the same time, as

f ( 6 + ar) - f ( 6 + ar) f(5 + ar) -f(S 1 2 1 a-u 1

2

+ ar) 3

7

a-a

2

1

3

i.e., F(a, a) 2 F(U, a). 1 2

1

(5.15)

Q.E.D.

3

For a given convex function f(4) on (5. and a given point 5 E 6,we choose a vector r E R” arbitrarily when 5 is not a boundary point of (5. and, when 6 is on the boundary of 6,in such a way that there exists a positive number h such that 5 + hr E 6. (We assume that 6 is neither empty nor a single point.) Then, for sufficiently small h (>O), F(h, 0) = Cf(6 + hr) -f(S)llh

(5.16)

is defined and is monotone nondecreasing as a function of h. Furthermore, we can find an h‘( > 0) such that 6 - h’r E ID, where ID means an open convex set mentioned in Definition 5.2. Then we have

F(h, -h’) = Cf(4 + hr)-g(5 - h‘r)l/(h F(0, A’) = Cf(9- s(5 - h’r)I/h’,

+ h’),

(5.17)

where g(6) is the convex function on D mentioned in Definition 5.2. From the preceding lemma, we have F(h, 0) 2 F(h, -h’) 2 F(0, A’),

(5.18)

so that F(h, 0) is bounded below by P(0, -A’) and, consequently, there exists lim F(h, 0). This limit is called the deriuatiue o f f ( 6 ) in the direction h++O

ofr and is denoted byf’(6; r). Thus we have Theorem 5.4. A convex function f(5) on a convex set 6 has, at every point 6 E 6,the derivative in the direction of every admissible r:

where r is called “admissible” when we can find an h(>O) such that 5 hr E 6.

+

Theorem 5.5. f’(5; r) of a convex functionf(6) on 6 has the following properties, where 5 is an arbitrary point in (5.:

(i)

f’(6;

0) = 0.

(5.20)

16

11. Convex Sets and Functions

(ii)

f’(5; ar) = af’(5;r) for every a (20) and every admissible r. (5.21)

(iii) If r and r are admissible, then so is r = (1 - 2)r A

1

0

0

+ k1 for any 1

between 0 and 1, and

(iv) If both r and -r are admissible, then

(v) then

(5.23) f’(5; r>2 -f’(5; -9. For a fixed r, if u B 0, p >= 0, a > p, 6 + c(r E 6,and 5 + pr E 6,

-f‘(5

+ ur; -r)

Zf’(5 + pr; r)

(5.24)

and

f’(5 + ar; r) Z f ‘ ( 5 + pr; r).

(5.25)

PROOF. (i) and (ii) are obvious from the definition off’(6; r). (iii) follows from the relation

f { 4 + hC(1 - 4 0r + Arll -f(U 1 5 (1 - 4Cf(S + hr) -f(5)1 0

+ X f ( S + hr) -f(S)1

by dividing by h and taking the limit h -+ +O; (iv) follows from (i) and (iii) because

W ( S ; r) + +f‘(S;

-r)Z f ’ ( 5 , O )

= 0.

If we take an h such that 0 < h < u - p, then, by Lemma 5.1, we have

f ( 5 + 4 -f(S + ar - hr) ,f(S + ur>-f(S a

- (a - h )

+ fir)

u-B 2

-

f(S + pr + hr) -f(5 (B + h) - B

+pr>. (5*26)

Taking the limit h --f +O, we obtain (5.24) of (v). (5.25) follows from (iv) and (5.24). Q.E.D.

17

5. Convex Functions

REMARK 5.1. If the function f(5) is partially differentiable in the ordinary sense, then

af f ' ( 5 ; r) = , =1I rK at"

(5.27)

Y

so that (i)-(iv) in Theorem 5.5 are reduced to the trivial consequences of the relation (5.28)

Iff( 5) is twice-differentiable, then

so that (v) is equivalent to the positive definiteness of the matrix [azf/atl atK]( K , A = 1 , ...,n). Finally we add a few words on the relations between convex functions and convex sets.

Theorem 5.6. If f(5) is a convex function defined on a convex set the set 6 ( ~ ( 5 : )of those points 5 for whichf(5) 6 0

(5: (sR"),then

6 = (515 E (5:,f(5)5 01

(5.30)

is a convex set that is relatively closed in (5:.

+ A5

PROOF. For any 5,5 E 6 s KandanyA(0 S A 5 l), 6 = (1 - A)k 0

belongs to

(5:

1

1

0

1

and

f(5) s (1 - am) + A f (5) 5 0, 1 0 1 so that

5 belongs to

G. Therefore, 6 is convex. Sincef(5) is continuous

1

(Theorem 5.3), 6 is closed relative to

(5:.

Q.E.D.

A linear function n

l(5) = I(=

1

aKt"

+ a,

(a,, a, : real numbers)

(5.31)

is obviously (not strictly) convex and, at the same time, concave. Hence, from Theorem 5.2, follows

Theorem 5.7. Iff(5) is a convex function (on 6)and l(5) is a linear

18

11. Convex Sets and Functions

function (on R“),

is a convex set relatively closed in

(5.

PROOF.f ( Q - f(5) is a convex function.

Q.E.D.

5.2. Convex Functions with One Variable In this section, we consider the special properties of convex functions of one variable. It is obvious how the properties of convex functions investigated in the preceding section are specialized in the case of convex functions with one variable. In particular, a convex function with one variable f(5) defined on an interval Z is continuous at every point of I , and it has both the right derivative

.f‘(+’(t) =f’(5;

+ 1)

(5.33)

and the left derivative

= -f’((;

f’(-’(()

- 1)

(5.34)

at every interior point 5 of I. (It has only the right or the left derivative if 5 is the left or the right end of Z,respectively, cf. Theorems 5.3 and 5.4.) By virtue of Theorem 5.5, f’“’(5) as well as f ’ ( - ’ ( < ) is monotone nondecreasing with 5, and, at every interior point 5 of I, f ” + ’(5) 2 f” - ’(5).

(5.35)

Furthermore,

(5.36) Theorem 5.8. For any convex function f ( 5 ) defined on I, there exist in Z at most a countable number of 5’s at which

>f”-’ (5).

ft{+)(<)

(5.37)

PROOF. Let the set of such 5’s be denoted by Eo. To each element of Eo , we make correspond an open interval J ( 5 ) : If

J ( 5 ) = {rllf’(+)(5) > rl >f“-’(5)). 5 # 5 (5, 5 E Eo), J ( 5 ) n J ( 5 ) = 0, due to (5.36). 1

2

1

2

1

5

(5.38) Since each J ( 5 )

2

contains at least one (in fact, countably infinite) rational number and the totality of real numbers R [ 2 U J ( 5 ) ] contains only a countable number C€Z0

5. Convex Functions

19

of rational numbers, Z,, cannot contain more than countably infinite elements. Q.E.D. We may represent a convex functionf(5) by a curve vl =f '(5)

(5.39)

on the plane with abscissa 5 and ordinate q, which curve we shall call hereafter the characteristic curue off(5). In more precise terms, it is defined as follows. Definition 5.3. The characteristic curve of a convex function f(5) defined on Z is the set C of points (5, q) such that 5 is an interior point of Z and f'(-)(c) 5 q S f ' ( + ) ( t )5, is the left end of Z and - 03 < q Sf'(+)(C), or 5 is the right end of Z andf'(-)(t) S q < co.

In what follows, we shall consider only such convex functions as are defined on " closed intervals," because it suffices for later applications. As is exemplified in Fig. 5.1, the characteristic curve C of a convex function f(5) defined on a closed interval Z = [b, c] [b may be - co and

Fig. 5.1. Characteristiccurve and the associated convex functions.

20

11. Convex Sets and Functions

where f " + ) ( c )or f ' ( - ) ( b )is assumed to be 00 or - CQ, respectively. Obviously, P + , P-, and C are mutually disjoint and together cover the whole (5, q) plane. Equations (5.35) and (5.36) may be rephrased as follows: (i) For any points (5, q ) E C and (5, q ) E C, 1

2

1

2

From (i), i.e., the monotonicity of C, and the fact that a point ( 5 , q ) sufficiently far away northwestward from the origin belongs to P+ and a point sufficiently far away southeastward from the origin belongs to Pfollows another property of C: (ii) For any real number I , the straight line ((5, $15 sects C at one and only one point.

+ q = I } inter-

Conversely, if a set of points C on the (t, q) plane satisfies both (i) and (ii), C is a monotone continuous curve on the plane dividing it into two parts P+ and P - . Furthermore, if we define the functionf(5) by the integral along the curve C from a fixed point (5, q) to a variable point 0

0

(t, vl), i.e., by 1

1

f(5) 1

=

f($.a,"';d5 + an integration constant,

(5.41)

thenf(5) is a convex function, as will be seen without difficulty. [Note that the existence of the integral (5.41) is also assured by (i) and (ii). See Remark 5.3 at the end of this section.] The characteristic curve of the functionf(5) defined by (5.41) coincides with the C itself. It will also be evident that two convex functions differ from each other only by an additive constant if and only if they have one and the same characteristic curve. Although we have adopted the assumption that the domains of convex functions to be dealt with are closed, their range may not be closed. In view of the symmetry of conditions (i) and (ii) with regard to 5 and q and for the purpose of making the descriptions to follow also symmetric, we

21

5. Convex Functions

shall impose the following condition on C,i.e., we shall restrict ourselves to consider only those functions whose characteristic curves satisfy (iii) : I

I

(iii) The projection of C to the ( axis as well as that to the q axis is a closed interval. From (iii), it immediately follows that (iv) If the domain I of a functionf((), i.e., the projection I of its characteristic curve C to the 5 axis, is unbounded to the right (to the left), then either sup f’(+)(()= co [inf,f’(-’(r) = -a] or <€I

5EI

there exists a finite ( such that, for all ( > 5 (all 0

f”+’(t) C f ’ ( 0 =f“-’(5)1. 0 0

5 < < ) , f ’ ( t )=

0

0

By virtue of the symmetry of (i), (ii), and (iii) with respect to ( and q, we can associate with a curve C, besides the functionf(5) in (5.41), also the convex function g(q) defined by the integral along C :

S($.a, 5 ( € 9

g(q) = 1

11)

dq

+ an integration constant.

(5.42)

f ( 5 ) and g(q) are defined, respectively, on the interval I , which is the projection of C to the 5 axis, and on the interval J, which is the projection of C to the q axis. In summary, we can associate with every curve C with the properties (i), (ii), (iii) a pair of convex functionsf((), g(q) (determined to within an additive constant) by the integrals of the form (5.41), (5.42), and, conversely, if a convex function f ( 5 ) or g(q) defined on a closed interval and with the property (iv) is given, then we can determine uniquely a curve C (its characteristic curve) with the properties (i), (ii), (iii) by differentiation by ( or q :

(5.43) The two convex functionsf(5) and g(q) connected with one and the same curve C in this way are said to be conjugate. Next, we shall obtain a relation that connects conjugate functions directly without resort to the characteristic curve. Let us consider a curve

22

IT. Convex Sets and Functions

C with properties (i), (ii), (iii) (whose projection to the 5 axis is denoted by I and that to the q axis by J ) , fix a point 0 = (5, q) E C. For an arbitrary point E = (5, q) with along C,

5EI

0 0

and

u] E J

(not necessarily on C), two integrals

(5.44.1) and (5.44.2) are defined. The value of (5.44.1) or (5.44.2) depends only on 5 or q, respectively, but not on the way of choosing the point B = (5, q) or 1

D = (l,9). (The ambiguity of choice takes place only when C has a vertical I

or horizontal segment.) In each of the four possible cases of the relative position of E to C, it will be seen, from property (i), that

= area

BDE 2 0,

(5.45)

and that the left-hand side of (5.45) is equal to zero when and only when the point E = (<, q) is on C (cf. Fig. 5.2). From (5.41), (5.42), and (5.49, we have, for any 5 E I and any q E J , .(<Jl)

.f(O + s(u])= J

I

‘$2)

(v

=5q-

- u l ) d5 0

5 q + c,

+

1v.2, ( 5 (5.v)

-5) 0

4 + q(5 - 0 + 5(q- 4 +c 0 0 0 0

(5.46)

00

where c is a constant not depending on or q . If it is understood that the integration constants in (5.41) and (5.42) are chosen in such a way that c = 5rl i n (5.46), then we have 00

Theorem 5.9. If a convex function f(5) defined on a closed interval 1 and a convex function g(q) defined on a closed interval J are mutually

23

5. Convex Functions

Fig. 5.2. Proof of the fundamental inequality.

conjugate, then for every

5 E Z and every rj E J,

(5.47) f ( 5 ) + drl) 2 5% the equality holding when and only when (5, y ~ ) is on the characteristic curve off(5) and g(q). As a direct consequence of Theorem 5.9, the following relations between a pair of conjugate functions are obtained:

drl)= maxCvS -f(01

(v E J > ,

(5.48.1)

(5 E 0.

(5.48.2)

<€I

f ( 0 = maxC5vr - dr)l qEJ

(5.48.1) and (5.48.2) may be regarded as explicit expressions of the transformation from a convex function to its conjugate and of its inverse transformation. It is remarkable that the transformation is of the same form as its inverse. This kind of transformation is called the “maximum transformation.” [The reader should show that, if q # J, the right-hand side of (5.48.1) is unbounded upwards, and similarly for (5.48.2).]

24

11. Convex Sets and Functions

The maximum transform, as is represented in the foregoing, can be generalized to define the conjugate pairs of convex functions with many variables :

[ max [

r'.

K=

1

rll....,rln

K=

1

g ( V l ~ . . - ~ V n ) = max ..., Cn

f(T',...,r")=

(5.49.1)

~KT"-f(t',***,r")],

~ " v ~ - s ( v I , . . . , v ~ ] , (5.49.2)

but we shall not go further into the details (see Bellman and Karush [I]).

REMARK5.2. We shall sometimes call a maximal subset of (more than one) points of C with the same q a horizontal segment and a maximal subset of (more than one) points of C with the same 5 a vertical segment. It is shown in Theorem 5.8 that the number of vertical segments is at most countable. Similarly, it can be shown that the number of horizontal segments is at most countable as well. REMARK 5.3. It seems necessary to show that the integrals along C, such as (5.41), (5.42), (5.44), etc., are legitimately defined. Let us take up the integral (5.41). We choose, on C, N + 1 points (5, fj), (5, ij), .. . , ( f

,

N-1

ij ), N-1

(t, V ) =(t,ij>, 0 0 0 0 and put

A(=

t

i

i+l

-t, i

Since N- 1

05

(5,V) 1

1

=(t,ii>, N N

i+l

i

c VAtS

N- 1

N-1

ij

At-

i=O i + l

i

i=O i

i

r6 r i

w5

9

i+l

i

c,

i+l

(i=O,...,N-l).

Aij= ij - i j i

1 1

0 0

(t, i j ) in such a way that N N

AijAfS(q-q)maxA[ i=O

i

1

i

0

i

i

there exists the limit

c

N- 1

lim

c ijAt,

N-1

i j A f = lim

N - + m i=O i + l

i

N + m i=O i

i

where it is understood that max At, as well as max Aij, tends to zero as i

N+

00.

i

i

i

This limit is, by definition, the integral in (5.41)

25

5. Convex Functions

By direct calculation, we can show the convexity of the function f ( 5 ) defined by (5.41), etc., from the properties (i), (ii), (iii) of C.

REMARK5.4. It can be shown that if and only if C is a straight line passing the origin,

f (5) = 9(?)

for every

(5, rl) E c.

The reader should try to verify this fact. Example of maximum transforms: (1)

f(0= *at2

[I = ( - 0 0 ,

W)l,

9(?) = (2a)-'v2

CJ

m>l,

= (-

00,

and

c = ((5, ?)I? = at, 5 E 1, ? E J > , where a > 0 (see Fig. 5.3).

Fig. 5.3. Conjugate functions: example (1).

26

11. Convex Sets and Functions

Fig. 5.4. Conjugate functions: example (2).

Fig. 5.5. Conjugate functions: example (3).

6. MINIMUM OF A CONVEX FUNCTION ON A CONVEX SET

Definition 6.1. A point in the n-dimensional Euclidean space R" is called a local minimum point (or, simply, local minimum) of a function f ( 6 ) defined on a subset 0: of R", if there exists a neighborhood U ( t ) of f

6. Minimum of a Convex Function on a Convex Set

27

5 E ~ ( tn)6.

(6.1)

such that

f(5)~

f ( t ) for every

t

If (6.1) holds for every 5 E 6, is called a global minimum point (or, simply, global minimum or minimum). When f (5) is a convex function defined on a convex set 6,we have the following important theorem. Theorem 6.1. A local minimum of a convex function f (5) on a convex set 6 is a global minimum, and the set of global minima is a convex set relatively closed in 6. If.f'(t) is strictly convex, there exists at most one global minimum.

PROOF. Let ( be a local minimum off(5) and U ( t ) be a neighborhood as mentioned in Definition 6.1. If is not a global minimum, then there should exist a point 4 E 6 such that f ( 5 ) < f ( t ) . Sincef(5) is convex,

t

belongs to 6 due to the convexity of 6. Furthermore, we can choose 1 so small that 5 may belong to U ( t ) .This, however, leads to a contradiction a

[cf. (6.1) and (6.2)]. Let the set of global minima be denoted by %R. For any 5 and 5 in !lX and any 1(0 5 15 l), we have 1

0

Since 5 as well as

5 is a global

minimum, it is impossible for f(5) to be

1

0

smaller thanf(5) = f ( k ) , which means that is.,

5 E %R,

1

0

for every

5 is also a a

A (0 5 15 1). Therefore,

a

global minimum,

%R is a convex set. The

A

relative closedness of !IN follows from the continuity of f(5). Finally, let us suppose that f ( 5 ) is strictly convex and that there exist two distinct points 5 and 5 in !lX. Then 0

1

f (112 5 ) < 3 0m + 3 M1 ) =f ( 05 ) =f ( 41 ) , which is absurd.

(6.5) Q.E.D.

I11 Topological Properties of Networks

The common feature of many kinds of network-type problems is their essential dependence on a geometrical figure called a linear graph or, simply, a graph. We shall give a summary of graph theory in this chapter. A graph is a figure consisting of lines and points so that its properties are easy to investigate intuitively. In the standard textbooks on graph theory (see, e.g., Konig [l], Berge [l], and Ore [l]), “points ” are made to play a principal role, whereas “lines” are merely a subset of the set of pairs of points. However, in applications, it is often the case that lines are endowed with more physical significance, whereas points are merely their junctures. Our standpoint, from which the following expositions will be made, is to pay attention primarily to branches, i.e., lines, taking account of nodes, i.e., points, only when it is necessary to do so (cf. Kondo and Iri [l] and ~ r c11, i ~41). 7. GRAPHS AND VECTOR SPACES

Unless otherwise stated, graphs to be treated in the following are “finite,” i.e., they consist of a finite number of nodes and branches. 7.1. Definition of a Graph Intuitively, a graph is a geometrical figure consisting of points (which we shall call nodes) and lines or line segments (which we shall call branches)

28

29

7. Graphs and Vector Spaces

connecting a node to another. In the formal terminology, it may be defined as follows. A graph 6 is the composite concept { 111, 23, a+, a-} of (a) a set of nodes 111 = {nl, . . ., nM}, (b) a set of branches 23 = { b,, ..., bn}, (c) an incidence relation, where the incidence relation is a couple of functions domain 23 and the range in 111:

a-:

a + : 23-111,

a+ and a-

with the

23-a.

We sometimes call the cxth element n, of 111 “node a’’ and the Kth element b, of 23 “ branch K.” The intuitive interpretation of the functions a+ and a- is that nodes na and nB are the two ends of branch b, , or, more precisely, b, starts from na and ends at nB if and only if a+b, = na and a-b, = nB (cf. Fig. 7.1). We say that b, is positively incident to d’b,, negatively incident to a- b, , and not incident to all the other nodes.

Fig. 7.1. A branch and its endnodes.

An incidence relation can be represented by other functions. In fact, let 6+ and 6- be functions whose domain is 111 and whose range is in the power set 2” of 23 (i.e., the set of all subsets of 23):

a+: 111+29,

6-:

111-+29,

and which satisfy the conditions 6 + n a n 6+nB = 6-n, n 6 - n B = @ (empty set)

for every pair na , nB and M

U 6+na= a= 1

u M

a= 1

6-na=B.

(E

111, cx # p),

(7. I)

30

111. Topological Properties of Networks

Then we can define d + and S'n,

3 b,

6-n,

3

a-

uniquely by the relations if and only if d+b, = n a , if and only if a-b,

b,

= n,.

(7.3)

C(7.1) assures the single-valuedness of a+, a-, and (7.2) implies that the domain of a', a- is the entire 23.1 Conversely, if ' 8 and a- are given, then we can define 6' and 6- by means of (7.3), which satisfy (7.1) and (7.2). S'n, and 6-n, are intuitively interpreted as the set of branches positively incident to norand the set of branches negatively incident to nor, respectively (see Fig. 7.2).

Fig. 7.2. Set of branches incident to a node.

Sometimes, the functions

r+- a - 6 + :

r- = a + s - :

9t+2% ,

%+2%,

which satisfy the conditions na E

r+np

if and only if

np E

T-n,,

are used to represent the incidence relation of a graph (where a+, 8- are regarded as the mappings from 2' to 2', which are naturally defined by mcans of the original a+, 8- from B to '3).If a set % and a couple of functions r+,r- are given, we can construct a graph in the foregoing sense by defining B as the set of pairs (n,, na) (n,, na E %) such that n, E r - n ,

or

na E r + n ,

and the incidence relation by d+(nar na)

= nu

and

d-(n,,

nap) = na.

But the converse is not always possible, i.e., we cannot represent a graph having so-called "parallel branches" (i.e., the branches b, b, . . . with the common endnodes a+ b, = d'b, = , d - b, = d - b, = * * .) in this way.

31

7. Graphs and Vector Spaces

A node that does not belong to 8'23 u 8-23, or, equivalently, a node n, such that 8 + n , = 6-n, = 0, is called an isolated node. Usually it is convenient to exclude the isolated nodes from %. A branch b, such that a+b, = 8-b, is called a self-loop, which is a branch represented by an arrow starting from a node and returning to the same node. b2,d 2 + , a,-} Two graphs 6, = {%,, b,, a,', a,-} and 6, = {g2, are said to be isomorphic if there is a one-to-one correspondence between g1and %, as well as between bl and b2 and if, under these correspondences, 8,' and 8,- are isomorphic with a2+ and d 2 - . In what follows, however, we often make no distinction among isomorphic graphs, regarding them merely as identical. Now we shall introduce several concepts of transformations of graphs. a2= {ill2 B2, , a2+, a,-} is said to be obtained from = {%,, B,, a,+, a,-} by reorientation of a branch, say b,, of b,,if 5Q2 = g1,b2= b l (here, again, we use the equality for a one-to-one correspondence) and

a2+b,

= a, - ,b,

,

a2+b, = al+b,,

a,-

b, = al+ b ,, ,

a2-b, = a,-b,

for all

IC

#

I C ~ .

(7.4)

Intuitively, 6, is obtained by reversing the direction of (the arrow representing) branch ,b, . The reorientation of more than one branch is defined in a similar manner. B2= {a,,b2,d 2 + , a,-} is said to be obtained from 6, = {Sl, B1, al+, a,-} by open-circuiting a branch, say b, of b,,if S2 = Sl, b2= 23, - {bKo},and a,+, 8,- are the restrictions of a,+, 8,- to B2, where {bKo} denotes the set consisting of the single element b,, and by the restriction to A' of a functionfdefined on A (A' E A ) we mean the function g defined on A' which has the same value as f has at every element of A'. Intuitively, G2 is obtained by simply removing a branch b,, from (lil. The open-circuiting of more than one branch is defined similarly. (tj2 = {g2, b 2 ,d 2 + , a,-} is said to be obtained from 8 , = {%,, b,, a,+, 8,- } by short-circuiting a branch, say ,b, , of bl,if they are related to each other as follows. consists of the nodes of %, which are not incident to b, and of a node not in 8, that we shall denote by n o . We denote by f a function such that

f:

%1+%2,

f(nJ = n,

f(',+

'KO)

= 1f '(

if na is not incident to b, - bKO) = "0 *

(7.5)

32

111. Topological Properties of Networks

B, = Bl - { bKo},and a2+b, =f(a,'b,),

for all

dz-b, =f(a,-b,)

IC

# q,. (7.6)

Intuitively, 6, is obtained by identifying, i.e., combining into one node, the two endnodes of b, and then removing bKo.The short-circuiting of more than one branch is defined similarly. 6, is called a subgraph of G1 if the former is obtained from the latter by open-circuiting some branches. The complement of a subgraph (lj2 of G1 is, by definition, the subgraph obtained from 61by open-circuiting all those branches that belong to G 2 . (We sometimes exclude from consideration the isolated nodes resulting by open-circuiting.) 1.2. Vector Spaces

Let G be the ring formed of all integers. For a given graph 6 = {%, B,

a', 8-} (% = {nl, .. . , nM}, B = {bl, ...,bn}), we denote by V(G, 8)the totality of linear forms 5, with 23 as the basis and G as the domain of coefficients :

For two arbitrary elements n

5= C

rb,

and

2

,=l1

1

of V(G, B),their sum

g=

n

C rb, "=12

5 + 5 is defined as 1

2

and for an arbitrary integer h (E G) and an arbitrary element

crb,

6=

n

E V(G,

B), their product hg is defined as

K = i

hS=

2

I(=

1

(h5")b"

CEV(G,~)l,

(7.9)

where, on the right-hand side of (7.8) or (7.9), 5" + 5" or hl" is the sum or 1

2

product of integers in G. Thus, V(G, B) may be regarded as a module or a vector space, so that we shall call an element of V(G, 23) a vector. Furthermore, we often make use of the function called inner product which maps V(G, B) x V(G, B)to G, i.e., we define the inner product of a

33

7. Graphs and Vector Spaces

vector

6=

c Fb, n

c q,b, n

23) and another q =

E V(G,

,=l

E V(G, 23) by

K=l

- 1 c t"?, n

5

(EG).

=

(7.10)

K= 1

We shall resort to the obvious abbreviations without particular remark. For example, we shall denote by 0 the vector with all = 0, by b,, the vector with all 5" = 0 except for tK0 = i 1, etc. According to this convention, the summation on the right-hand side of (7.7) could be identified with that defined in V(G, 23) such as (7.8) and the product in each term under the summation sign of (7.7) with that defined between G and V(G, 23) such as (7.9). Similarly, we define the vector space V(G, %) as the totality of the vectors of the form

c Cana M

4=

(7.11)

(Ca

a= 1

in which the sum of vectors and the product of a vector with an integer are defined as before. Instead of G, we can adopt another commutative ring, with the unit element and without divisors of the zero, to form a vector space. Especially, we shall deal with V(R, 23) and V(R, 8)in the later chapters (where R is the field of real numbers). Since the results in the case of integer coefficients can easily and naturally be adapted to the case of other coefficients, we shall consider, in this chapter, mostly the case of integer coefficients. The incidence relation of a graph 6 = {a, 23, a+, 8-} induces two homomorphisms 8 and 6 between V(G, 23) and V(G, %):

a:

V(G, 23) + V(G, %),

6: V(G, %) + V(G, b),

defined by the basic relations: ab, = a+b, - a-b,,

6na =

c

b,

c

-

b, ~ d .n +

Now let us define a matrix incidence matrix, by

6," (a= 1, ..., M;

bn E 8 -

K =

(7.12)

1, ..., n), called the

if

na = a+b, #a-b,

= -1

if

na=8-b,#a+b,

=O

otherwise,

D,"=1

b,. n.

(7.13)

where M is the number of nodes and n the number of branches. It is

34

111. Topological Properties of Networks

obvious that, for a fixed K, the number of nonnull 6,”s is either 0 or 2, that it is 0 if and only if b, is a self-loop, and that, otherwise, one of the nonnull 6,”s is equal to I and the other to - 1. Hence, we have =0 a=

In terms of

D:,

for all K.

(7.14)

I

(7.12) is written as (7.12‘)

or, in general, we have the explicit expressions for d and 6:

Thus,

a

and 6 are mutually “contragredient.” Direct calculation will

show that, for arbitrary 5 =

n

M

2 <“b,

E

V(G, 23) and

,=I

& = C lanu E V(G, %), a= 1

we have “

M

(7.16)

In particular, we have (db,)

- nu = b,

*

-

(6na) = D,“.

(7.17)

It should be noted that we can derive the incidence relation of a graph

Fig. 7.3. Example of a graph.

35

7. Graphs and Vector Spaces

from a, 6, or 6,' if there is no self-loop. (It cannot be known from and 6; to which node a self-loop is incident.)

a,

6,

EXAMPLE.Let us illustrate the concepts thus far introduced in connection with the graph of Fig. 7.3, where n, is an isolated node and b4 a self-loop. (A4 = 8).

~={nl,n2,n3,n4,n5,n6,n7,n,}

~3{b,,b2,b3,b4,b5,b6,b7,bs,bg,blo,bll}

-

D," =

1 2 3 1 1 0 1 2 - 1 - 1 1 0 0 - 1 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 8

4 5 6 0 - 1 1 0 0 0 0 1 - 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(n=11>-

7 8 9 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 - 1 - 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 - 1 0 0 0 0 0

36

111. Topological Properties of Networks

Finally, let us introduce some notations concerning vectors. To a vector n

4 = K = l ("b,

we make correspond the following three subsets of 23: car+ 5 = {b,ltK > 0},

5 = {b,lt" < 0}, 5 u car- 5,

car-

car 5 E {b,ltK # 0} = car'

(7.18)

which will be called the positive carrier, the negative currier, and the currier of 5, respectively. Obviously, car+ 5 n car-

5 = @.

Similar notations are used in connection with vectors of V(G, %), and with those of the vector spaces with real coefficients. By a simple vector in V(G, 23) or V(R, B), we mean a vector for which every t" is either 1, - 1, or 0. Similar definitions are used for simple vectors in V(G, %) or V(R, %).

7.3. Loops and Cutsets We call a vector

5 E V(G, 23) an (integral) loop if a6 = 0.

(7.19)

The totality of loops forms a linear subspace (or submodule) X(G) of V(G, a),i.e., for arbitrary 5 and 5 in X(G) and arbitrary integers h and h, 1

h5 11

+ h5 also belongs to X(G).

2

1

2

22

We call a vector q E V(G, 23) an (integral) cutset if there exists a vector 5 E V(G, %) such that q = sg.

(7.20)

The totality of cutsets forms another linear subspace Y(G) of V(G, 23). From (7.16) it immediately follows that the inner product of a loop and a cutset vanishes:

5* q =0

C5 E WG),

q E Y(G)l.

(7.21)

A loop with coefficients (,'s equal to 1, - 1, and 0 is called a simple loop, and a cutset with coefficients q,'s equal to 1, - I , and 0 a simple cutset. A subset 23' of 23 is said to be loop-free if there is no nonnull loop 5 ( # 0) such that car 5 23'. A subset 23' of 23 is said to be cutset-free if there is no nonnull cutset q ( # O ) such that car q 5 23'.

37

8. Connectivity

A minimal loop means a nonnull loop 6 such that every proper subset of car 6 is loop-free, and a minimal cutset means a nonnull cutset q such that every proper subset of car q is cutset-free. A simple minimal loop or cutset is called an elementary loop or elementary cutset, respectively. The intuitive meanings of loops and cutsets as well as their roles in graph theory will be discussed in $9. 8. CONNECTIVITY

To begin with, we define a path of length 1 from node nao to n, as the composite concept of a sequence of 1 + 1 nodes (nao, n,, . . . , n,,-, , n,) and a sequence of 1branches with signs (cl b,, , . . . , cl bK,), where c i = 1 or - 1, such that, for every i (= 1, . . . , l), nmi- = a+ b,,

(when ei = - l),

(when E , = 1)

or

8- b,

nOi = a - b K i (when ci= 1)

or

a + b K i (when ci = - 1).

(8.1)

Nodes nai's are said to lie on the path. Branch bKiis said to lie on the path in the positive direction or in the negative direction according to whether c i = 1 or - 1. In particular, a single node (and an empty sequence of branches) is regarded as a path of length 0 from that node to itself. If, for every i andJ (0 i < j 5 I, j - i < I), nai # naj , the path is called elementary. It will easily be seen that, from a given nonelementary path from na to nB, we can derive an elementary one (of shorter length) from na to nB by removing appropriate subsequences from the given sequences. Obviously we have bKi # b K j (i # j ) in an elementary path. The intuitive meaning of a path or elementary path will be evident from the very definition. Two nodes n, and ns are said to be mutually connected and denoted by "a

-

(8.2)

"B

when there exists a path from n. to n B . Evidently, the relation

--

--

reflexive, i.e., n, n, for every a, symmetric, i.e., if n, nB , then nB na, transitive, i.e., if na nB and np ny, then n,

N

" N "

is

n,,

so that it is an equivalence relation. The set of nodes '3 of a graph 6 is classified into the equivalence classes W,,. . . , !TIM-,,, by the relation "-," where M is the number of

38

111. Topological Properties of Networks

nodes and 0 5 m 5 M - 1: M -m

%=

u !Ri

i= I

(Wingj=@, i z j ) .

(8.3)

Two nodes are mutually connected if and only if they belong to the same class mi.I n accordance with the classification of W, the set of branches 8 is also classified, i.e., 8 is classified into dl, . . . , 23M-msuch that

u

u di

M -m

8,=

8=

(ij+nuu6-n,),

(Binsj=@,i # j ) .

i= 1

n, E 91,

(8.4)

For a given graph 8 = {%, 8, (3+, d - } , we can construct M - m graphs 6 ,= {92,, 23,,d i + , (3,-} (i = I , . . . , M - m), where 111, and Bi are as defined previously, and di+ and di- are the restrictions of 8' and 8- to %,, respectively. 8 , ' s are called the connected components of 6. 6 is said to be connected when M - m = 1 and unconnected when M - m 2 2. It is also evident that (7.14) can be generalized into

1

n, E 91i

foreveryx

DKa=O

Theorem 8.1. If nu

-

(i=l,

..., M - m ) .

n p , there exists a vector

(8.5)

6 E V(G, 8) such

that

nu - np = 86,

and vice versa.

-

PROOF. If nu = n p , (8.6) holds with 6 = 0. If nu np and nu # n p , there exists a path from nu to np with a sequence of nodes and a sequence of branches with signs which satisfy (8.1), where nu = nuo and np = n u , . Set

Then we have (8.6) from (8.1). Conversely, if a vector n

V(G, '3) is expressed as

5 = 36 with some 6 =

so that we have, from (8.5),

1

c"=O

for

i=1,

M

YnaE a= 1

1 t K b KE V(G, d),then K = 1

n, E 9li

5=

..., M - m .

39

8. Connectivity

Therefore, if na and n p satisfy (8.6) with some 5, they are mutually connected, i.e., belong to the same connected component; for, otherwise, 4 = na - n4 would not satisfy (8.8). Q.E.D. Let us select, arbitrarily, one node out of each connected component and denote them by n,,, . . . ,naM-, .To every vector 5 =

a= 1

M-m

we make correspond

M

1 p n a E V(G, %),

F = 1 p n a i , where i=l

Since (8.10) and since na and nai in each term of the right-hand side of (8.10) are mutually connected, there exists a 5 such that a

aat = na - nai

for every a.

Hence, we have

5 = z: + at,

(8.1 1)

with (8.12) The totality of the elements of V(G, %) which can be expressed in the form a t with some 6 in V(G, b)forms a linear subspace F(G) of V(G, %), M-m-.

and the totality of the elements of V(G, %) which are of the form

['nai

i= 1

also forms a linear subspace E(G). (8.11) shows that every vector of V(G, %) can be decomposed as the sum of a vector of E(G) and a vector 86 of F(G). Moreover, the decomposition is unique, for, if

5 = < + ag = g' + ag', then <'- g = ag - ag' = a(g

- g').

(8.13)

The left-hand side of (8.13) belongs to E(G), whereas the right-hand side

40

111. Topological Properties of Networks

belongs to F(G). Every vector of F(G) must satisfy the relation of the type (8.8), whereas it is evident that no nonnull vector of E(G) can satisfy the relation of the same type. Therefore, (8.13) must identically vanish, which guarantees the uniqueness of the decomposition. Thus, we have shown that V(G, %) is decomposed as the direct sum of E(G) and F(G). The dimension of V(G, 3)is M (the number of nodes), the dimension of E(G) is M - m (the number of connected components), and, since a vector of F(G) is of the form

(8.14) with F's arbitrary, the dimension of F(G) should be equal to the rank of the incidence matrix D,". Hence, we have Theorem 8.2. The number of nodes M of a graph is equal to the sum of the rank of the incidence matrix m and the number of connected components M - m.

EXAMPLE.The graph of Fig. 7.3 has three connected components, as shown in Fig. 8.1, i.e., we have

m = M - 3 = 8 - 3 = 5;

M-m=3,

% = { n l , ..., n51, %z = i n 6 9 n,>,

Bl = {bl, ' * * b 9 1 ; 2% = {bm b111; Bj3 = 9

9

a.

% = {nd,

We have known M - m dependence relations (8.5) to be satisfied by As in the foregoing, let us fix the set of nodes { n u , , . . . , naM- ,}, chosen one from each connected component. If we delete from the matrix 6; (with row index CI and column index K) those M - m rows which correspond to the nodes n u , , .. ,, denoting the resulting matrix with M - ( M - m) = m rows by

6,' whose rank is m.

B,"

(u=l,

.. . , m ;

~ = l ,. . ., n),

then the row vectors of B," are mutually linearly independent, and any row of 8; can be expressed as a linear combination of rows of D,". In fact, a row of 6,' is equal either to a row of B$ or to the negative of the sum of certain rows of D,",i.e., there exists a matrix A," consisting of 0, 1, and - 1 only such that m

D,"= 1 A,"B$ a= 1

(CI= 1 ,... , M ;

K =

1 ,..., n).

(8.15)

9. Loops, Cutsets, Trees, and Cotrees

41

(6,

Fig. 8.1. Connected components.

We shall call B," the reduced incidence matrix, which is determined depending on the choice of the nodes {n,,, . . , nUM-,). To decompose a given graph into the connected component, the following simple algorithm is of practical use. We assign the labels fa to the nodes nu, changing them step by step according to the following rules:

.

(i) Choose an arbitrary node n a , , and put f,,= 1 and f, = 0 for all a # al. (ii) Search for a node nu such thatf, = 0 and that there exists another node np E T'n, u T-n, (cf. $7.1) whose label fs= 1. If such an nu is found, then change the value off, from 0 to 1.

As (ii) is repeated, the number of nodes with label 1 monotonely increases until no node can be found any longer which has the desired property in (ii). Then, the set of nodes S1with label 1, together with the set of the branches B1 incident to them, constitutes a connected component 8 , containing n u , . By applying the same operation to % and 23 - B1, we obtain G2('iR2,?&), etc. 9. LOOPS, CUTSETS, TREES, AND COTREES

This section is devoted to the detailed investigation into the properties of loops and cutsets. The interrelations among loops and cutsets as well as their intuitive meanings will be clarified, and the way of constructing

42

111. Topological Properties of Networks

the basis for the subspace X(G) of loops and that for the subspace Y(G) of cutsets will explicitly be described.

9.1. Minimal Loops and Minimal Cutsets A loop or cutset with real coefficients, i.e., a loop EX(R) or cutset EY(R), is the equisignum sum of minimal ones, as will be shown in Appendix 2. An integral loop or cutset [EX(G) or EY(G)] may be regarded as a vector in X(R) or Y(R), so that it is decomposed as the equisignum sum of minimal ones in X(R) or Y(R). Multiplying by the least common multiple of the denominators of the rational numbers concerned, we have the equisignum decomposition of an integral multiple of the given integral loop or cutset into minimal integral loops or cutsets. Actually, an integral loop or cutset is decomposed as the equisignum sum of elementary ones (cf. Theorems 9.1 and 9.2 and Appendix 2). Thus, it is in due order to begin with minimal loops and minimal cutsets. A path from a node to the same node is called a closed path. If ( n a o , n,,, . . ., nar- , , nar = nao) and (ElbK,, . . ., b,) are the sequences of nodes and of branches defining a closed path, then a vector (9.1) is an integral loop [cf. (8.1)]. By definition, the sequence of branches of an elementary closed path does not contain that of another closed path as a proper subsequence. The following algorithm will decide whether there exists a closed path, the branches of which are included in a given subset of branches 23’= { bKl, . . . , b K l } ( E a), where the Jia’s are labels assigned to each node: (i) To begin with, put fb, = CI for all CI (= 1, ..., M ) , and put i = 0. (ii) Increase i by 1. Let the two ends of bKl be nai and nBi. If f i - l , a ,# f i - l , p , (we may assume f i - l , a ,< f i - l , p , without loss in generality), then put f 1.01 . =f. 1-1.q =fi--l,a

if

fi-l,a

=fi-l.Pi,

otherwise

and return to the beginning of (ii). If f i - l , a i

=fi--l,Bi,

then stop.

If (ii) is repeated until i attains I without stopping, then the nonexistence of the desired closed path is asserted. Since there is a path from nu to nB consisting of some of the branches b,, , .. . , bKi if and only if fi, ,=fi,as is

9. Loops, Cutsets, Trees, and Cotrees

43

easily verified, there exists a closed path consisting of b, and some of the branches b, , ..., bKi-l if and only if h-l,a, =fi-l, Bi for the two ends n,, and nBi of bKl. n

5 = 1 t"b, be a minimal loop. Without assume that car' 6 is nonempty. Take one branch, Let

loss in generality, we

W = l

and denote nao = a'b,

86 = 0

and n,,

and

= a-bKl.

a(<"'b,,)

=

say bKl, out of car'

6,

Since

rlnao - rK'nal

(5"' > 0),

there should exist-so long as na0 # nal-another branch bKzE car' 5 (or (or =d-b,,). Put naz = d-bKz(or =8'b,J. car- 6) such that nal = d'b, Then, either naz = nao or n,, = n,, or there exists another bKj E car' 5 (or (or =a-b,,). Continuing in this way, we car- 6) such that naz = d b' , shall finally get to an elementary closed path with the sequence of nodes (naF-, , . . ., nal = n a p - , ) and the sequence of branches ( E , bKr, . . ., el bKl) (1

5 r 5 0. If { bKr, .. ., b,J

1

c car

6 or 5 # 5 (=E, F r,Z zi b,),

then

6-5

t=r

is a nonnull loop whose carrier is a proper subset of car 6 [since car (5 - 5) does not contain bJ. Therefore, from the minimality of 6, 1

it follows that r = 1, car

6 = {b,,, . . . , bKz},and 5 = yX1 1 zi b,,.

Con-

i= 1

versely, let an elementary closed path be given with the sequence of nodes (nao, ..., rial) and the sequence of branches ( E b~, , ..., el b,J. Then, for arbitrary h,

is a minimal loop. That it is a loop is obvious. If there exists a minimal loop 5 such that car 5 c car 6,then the elementary closed path constructed from 5 in the manner just described would be properly included in the given closed path in contradiction to its elementariness. Thus, we have Theorem 9.1. A minimal loop is a multiple of the elementary loop corresponding to an elementary closed path. We have shown also that the intuitive image of an elementary loop is an elementary closed path. Next, we take up a minimal cutset. Consider a subset a' of the set and '$ -I 8'.A set of branof nodes % of a graph. % is dissected into 'ill' ches with signs {el b, , . .., bKl} (zi = 1 or - 1) is called a segregation

44

111. Topological Properties of Networks

associated with the dissection (%’, % - a’) if each bKi has one of its ends in %‘ and the other in % - 92’: %’

and

3- bKzE %‘

and

a+b,,

E

a- b, a+ b,

E

% - %’

when ci = 1,

E

% - %’

when ci = - 1,

and if no other branch has this property. In other words, if we opencircuit the branches of a (nonempty) segregation, the resulting graph has more connected components than the original. Conversely, let us suppose that the open-circuiting of the branches of 23’ ( ~ 2 3 )makes the number of connected components increase. For example, let us suppose that the node set %, of a connected component 8 , is disconnected into %,‘ and %, - %,‘. Then, 23’ obviously includes the segregation associated with the dissection (a,’, % - %,’). Since the open-circuiting does, in general, not decrease the number of connected components, we can discern whether or not a given subset 23‘ of branches contains a segregation by opencircuiting the branches of 23’ one by one and examining each time the number of connected components (cf. $8). Corresponding to a segregation { E , b,, . . . , c l bKl} associated with rhe dissection (%’, % - %’), a simple cutset

is defined. In fact, if we set

6=

CR,na n,

CEV(G,

(9.4)

%)I9

E

then q = Sl,.To prove this, we consider the coefficient of a branch b, in Sl,.If a+ b, E %’, b, appears in S(a+ b), with coefficient 1, and if a- b, E %’, it appears in S(a- b), with coefficient - 1. Therefore, if the two ends of b, belong both to %‘ or both to % - %’, car Sl, $ b, . In the remaining case where one of the two ends of b, belongs to %‘ and the other to % - %‘, i.e., where b, belongs to the segregation, car Sl, 3 b,. More precisely, if 3+b, E %‘ and 3-b, E % b, appears in S c with coefficient 1, and if a- b, E %‘ and a+ b, E % - %’, it appears in 66 with coefficient - 1. Hence, it should be that q = Sl,. A segregation is called an elementary segregation if there exists no other (nonempty) segregation included in it. %I,

M

n

Let q =

v],

b, be a minimal cutset. There exists a

K=l

V(G, %) such that q

= Sl,.

Let {el b,(,

. ..,

l, = a=

b,,}

1

lana E

be the segregation

45

9. Loops, Cutsets, Trees, and Cotrees

associated with the dissection (car'

5, ill - car'

c), and set

Then ij is a cutset, as we have seen. Moreover, car ij g car q.[For, consider b, E car ij. nu = d'b, must belong to car' 5 and nB = d r b, to ill - car' 5. Since q, = & (C, - Cp), C, > 0, and Cp 5 0, we have + q , > 0, which implies that b, E car q.] Thereforeq - ij is a cutset, and car(q - ij)c car q, since car(q - ij)$ b, . From the minimality of q, it follows that q = 5. We have shown also that the foregoing segregation {el bKl, .. . , el bKl}is elementary, since, otherwise, we should be led to a contradiction to the minimality of q. Conversely, the simple cutset q corresponding to an elementary segregation is minimal. For otherwise, there would exist a minimal cutset ij such that car ij c car q, and the segregation constructed from ij in the manner just described would be a proper subset of the given elementary segregation, which is absurd. Theorem 9.2. A minimal cutset is a multiple of the elementary cutset corresponding to an elementary segregation.

We have seen also that the intuitive image of an elementary cutset is an elementary segregation.

EXAMPLE.In the graph of Fig. 7.3, 6 = b, + b3 - b, is a loop [see Fig. 9.l(a)]. In fact, 86 = db, + db, - db, = (n, - n,) + (n2 - n3) (n, - n3) = 0. q = b3 - b, b, - b, is a cutset [see Fig. 9.l(b)]. In fact, if we take = n, + n, + n4 n,, we have 6c = an, + 6n, + 6n, + 6n,

+

c

+ + + b,j - b7) + (-b, - b2 + b3) + (b,

= (b, b, - b, b, - b, + b, - b, = q. It is verified that

6 * q = 0.

- b,

- b9) + b9 =

9.2. Rank and Nullity of a Graph The rank of a graph is defined to be the dimension of Y(G) [or, equivalently, of Y(R)], i.e., the number of linearly independent cutsets. The nullity of a graph is the dimension of X(G) [or X(R)], i.e., the number of linearly independent loops. n

Since a cutset q =

C q, b, I(=

1

is defined by

46

111. Topological Properties of Networks "2

(b)

Fig. 9.1. (a) Loop; (b) cutset.

with arbitrary

6,

the rank of a graph coincides with the rank m of its n

incidence matrix

DK3.Since a loop 6 =

1 ("bK is defined as a solution of K = l

the nullity of a graph coincides with the number of linearly independent

47

9. Loops, Cutsets, Trees, and Cotrees

solutions of (9.7), i.e., it is equal to n - rn (which will be denoted by k in what follows). Thus we have Theorem 9.3. The number n of branches of a graph is equal to the sum of its rank m and the nullity k. The orthogonality relation (7.21) between a loop and a cutset can be strengthened as follows. Theorem 9.4. The inner product 6 * q of a loop 6 [EX(G)] and a cutset q [E Y(G)] is equal to 0. Conversely, a vector 6 E V(G, 23) is a loop, if the inner product with an arbitrary cutset vanishes, and a vector q EV(G, 23) is a cutset, if the inner product with an arbitrary loop vanishes. PROOF. It remains to prove the second half of the theorem. Suppose that the inner product of a given 6 E V(G, 23) and an arbitrary cutset q E Y(G) vanishes. A cutset q is expressed as q = 86 so that we have, by virtue of (7.16), 0 = 6 * q = 6 * (86) = (86) Since

6

is arbitrary, 86 should vanish, i.e.,

- 5.

(9.8)

6 should

be a loop. Next,

q K b, E V(G, 23) and an

suppose that the inner product of a given q = K = l

arbitrary loop

6 E X(G)

vanishes. We determine

5=

M

lanu as follows. a= 1

Choose M - rn nodes nu,, . . . , nu,-,,, one from each connected component, i.e., nai E W i(i = 1, . . ., M - m),and put

rai=O

( i = 1, ..., M - r n ) .

(9.9)

For every node na other than nmi’s,there exists a path from it to one of ~ ...,E~ b,J. nai’s, of which the sequence of branches with signs is ( E b,,, Then, set (9.10)

We observe that the value of la is determined independently of the choice of the path from no to one of nai’s. For, if there is another path ( E l ’ b,,, . . .,Eh’bA,,) and

rl=

h

E i ? A ~ , j= 1

(9.11)

48

111. Topological Properties of Networks

then (elbKi, ..., el bKt, -eh'bAh, branches of a closed path, and

. . . , -cl'bA,)

is obviously the sequence of

h

1

(9.12) is a loop. We have, by assumption, (9.13) n

so that 5,' = l a . Let us consider 'the cutset Tj =

C 4, b,

= Sc

with

5

K=l

thus determined. If, for an arbitrary branch, we denote na = a'b, nB = 8-b,, we have

V,

= bK

*

(65) = ( n a - nfi)

*

5 = 5a -

*

and (9.14)

If (cl bKl, . . . , el bKJ is (the sequence of branches of) a path from nB to an n a i , then (b,, el bKl, . . ., el bKJis a path from nu to n a i , so that we have [a = VK

Therefore, ij, = q K for every shown that q is a cutset.

K.

+ 58.

Hence, q = T j

(9.15) = 85.

Thus, it has been Q.E.D.

9.3. Trees and Cotrees

A tree 2 on a graph 0 is a maximal loop-free subset of the branch set 8 of 6,i.e., 2 ( s 23) is a tree if and only if 2 is loop-free, and no loop-free subset of 23 strictly includes 2. Similarly, a cotree on a graph 6 is a maximal cutset-free subset of 23. As was shown in $9.1, for 2 to be loop-free it is necessary and sufficient that there is no closed path consisting of branches of 2 only, and for T to be cutset-free it is necessary and sufficient that there is no segregation consisting of branches of z only. It is also noted that the nonexistence of a segregation consisting of branches of % is equivalent to the existence of a path between any pair of nodes (belonging to the same connected component) which consists of branches of 23 - only. The practical algorithm for examining the loop-freeness or cutsetfreeness of a subset of branches was already given in $9.1. Thus, to obtain which is oba tree we may proceed as follows. We start from To = 0, viously loop-free. If there is a branch b, E 23 - ZiP1such that u {b,} remains loop-free, then we put 2,=Z i - l u {b,}. Repeating in this way

49

9. Loops, Cutsets, Trees, and Cotrees

.

for i = 1, 2, . ., we finally get to a maximal loop-free subset 2, with some E, which is a tree. A cotree can be obtained in a similar way. Theorem 9.5. For any given loop-free subset of 23, we can find a tree including it. For any given cutset-free subset of 8,we can find a cotree including it. PROOF, This is obvious from the definition. Q.E.D. Let 2 be a tree. For an arbitrary b, that is not in 2,2 u { b}, loop-free any longer. Consequently, there is a loop E, such that b,

(b, E 23 - 2).

E, E 2 u {b,}

E car

is not (9.16)

By virtue of Theorem 9.1, we may assume that E, is an elementary loop, i.e.,

E, = b,

+C

(tA= 0,1, or - 1).

tabb,

(9.17)

b i s l

Every nonnull loop whose carrier is in 2 u { b,} is a multiple of E,. For, if

is such a loop,

-

1

5 - pE,=

( f A - pta)b,

brsI

is also a loop whose carrier is in 2,so that it must be the null vector. Hence, corresponding to each branch in 23 - 2,an elementary loop of the form (9.17) is uniquely determined. Let us denote the number of branches of b - 2 by E and the set of those elementary loops by n

rp =

1 R,Kb,

( p = 1, ..., E),

(9.18)

K=l

where RpKis 1, 0, or - 1. If we renumber the branches in such a way that the first E branches constitute b - 2 and the remaining n - ]; branches constitute 2 and if we denote the renumbered set of branches by {vl, ..., vE,ul, . .,u,,-~}(={b,, ... , b,,}), then (9.18) is written as follows [cf. (9.17)] :

.

n-i

rp = vp

+ 1 M/u,

( p = I,...,

Q,

a=l

where M,"

= 0,

1, or - 1. Every loop f

c t"b,= c n

5=

tC=l

p= 1

xpvp

+

n-i

gaua a= 1

(9.18')

50

111. Topological Properties of Networks

is expressed as a linear combination of rp's: k

t = 1 xPrp

r" =

or

r;

RpKxP.

(9.19)

p= 1

p= 1

For r;

k

5-

1 xPrp= 1 (g a - 1 M / x P)u,

p= 1

a n= - k1

p=l

(9.20)

is a loop whose carrier is in 2,which must be the null vector. As is obvious from expression (9. IS'), the rp)s are linearly independent, from which it follows also that the expression of an arbitrary loop as the linear combination of rp)s is unique. Hence, I; is equal to the nullity k of the graph and n - E to the rank m. We shall call {rl, . . . , rk} defined as in the foregoing the primitive (or fundamental) set of loops associated with tree 2,and R," ( p = 1, . . . , k ; K = 1, . . . , n) the primitive loop matrix associated with 2. In entirely the same way, for a cotree 2,the elementary cutset whose carrier consists of a branch in 8 - Z and some branches of -L is uniquely determined. By renumbering branches, we assume that {ul, . . . , u i } form 23 - T, and {v,, . . ., v ~ - form ~ } Z ({ul, . . . , uiii, vl, . . . , v , , - ~ }= { b,, . . ., b,,}). Then, the primitive (or .fundamental) set of cutsets associated with Z, {d', . . . , a"}, is determined: n-iH

d" = u,

+ 1 N,"v, p= 1

n

=

C K=

DKabK

( a = 1 ,..., E),

(9.21)

1

where Npa, D," = 0, 1, or - 1. D," is called the primitive cutset matrix associated with T. An arbitrary cutset q is expressed, uniquely, as a linear combination of da's and iii is equal to the rank m of the graph and n - iii to the nullity k . The following theorem is obvious from the preceding considerations. Theorem 9.6. The number of branches of a tree on a graph is equal to the rank m of the graph, and that of branches of a cotree to the nullity k. Covollavy 9.1. A loop-free subset of m branches is a tree, and a cutsetfree subset of k branches is a cotree. PROOF. I t follows from Theorems 9.5 and 9.6. Q.E.D.

Not only is the sum of the number of branches of a tree and that of a cotree equal to the total number of branches (Theorems 9.3 and 9.6), but also we have

51

9. Loops, Cutsets, Trees, and Cotrees

Theorem 9.7. If 2 is a tree, then 23 - 2 is a cotree. If Z is a cotree, then 23 - T is a tree.

PROOF. 23 - 2 is cutset-free. For, if q is a cutset whose carrier is included in 23 - 2,i.e., if k

q=

c

(9.22)

YpVp,

p= 1

where the vp)s are branches of 23 - 2,then q must be orthogonal with every loop of the primitive set associated with 2 [cf. (9.1 S')] : rp-q=yp=O

( p = 1, ..., k),

(9.23)

so that q must be null. Since 2 has m branches, 23 - 2 has k branches and, consequently, is a cotree (Corollary 9.1). The latter half of the theorem is proved in the same way. Q.E.D. Theorem 9.7 states that a tree or cotree has always its complementary cotree or tree. Hence, in what follows, we shall always consider a dissection of 23 into a tree-cotree pair (2, 2) instead of a tree or a cotree alone. EXAMPLE.On the graph of Fig. 7.3, {bl, b5, b,, b, bll} is a tree, and { b, , b, , b, b, , b, , blo} is its cotree (see Fig. 9.2). The associated primitive loop matrix is 1

RpK=

1 2 3 4

1 0

0 0 0

65 1

2 1

3 1

0 0 0 0 0

0 1 0 0 0 0

4 0 0 1 0 00

5 0 1 0 1 1 0

6 0 0 0 1 0 0

7 0 0 0 0 1 0

8 0 0 0 0 1 0

9 1 0 1 1 0 0 0 0 0 0 0 0 0 , (9.24) 0 0 0 0 0 0 0 1 1

and the associated primitive cutset matrix is

1 2 D," = 3 4 5

1 1 0 0 0 0

2 3 4 5 6 1 1 0 0 0 0 - 1 0 1 - 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 0 0 1 0 0

8 0 1 1 0 0

9 1 0 1 1 0 0 0 0 0 0 0 0 0' 1 0 0 0 - 1 1 (9.25)

52

111. Topological Properties of Networks

Fig. 9.2. Tree and cotree.

The matrices M,," in (9.18') and N,," in (9.21) are, respectively, a\

1

p\

1 2 Mp"= 3 4 5 6

-

2

1 1 0 0 0 0

0 1 0 1 1 0

3 0 0 0 0

4 0 0 0 0

1

0

0

0

5 0 0 0 0 0 1

(9.26)

and )
3 4 5 0 0 0 0 -1 1 0 0 - 1 0 0 0 0 0 0 -

6 0 0 0' 0 1

(9.27)

where b1,u 2 , u 3 , u4, u5)=(b1, b 5 , b,, b, bll) and (vl, v2, v 3 , v,, v 5 , v6) = (b2, b3 , b, b, , b, , blo). The primitive set of loops or that of cutsets is given by (9.18), (9.18'), or (9.21) with RpKand MPa or D," and Npashown in (9.24)-(9.27). It follows from the orthogonality of a loop and a cutset that rp * d" = 0

for all p and a,

(9.28)

9. Loops, Cutsets, Trees, and Cotrees

53

or, in terms of the matrices R," and D,",

1 RpKDKa= 0

( p = 1, ..., k ; a

=

1,..., m),

(9.29)

K = l

where the tree-cotree pair with which rp's and RPKare associated may not necessarily be identical with that for d"'s and D,". If the associated treecotree pair is the same for all of them, (9.28) or (9.29) is rewritten, in terms of the matrices M," and Npa,as follows: Npa

+ Mpa= 0

( p = 1, . . ., k ; u = 1,

. . ., WZ).

(9.30)

[In the foregoing example the relation (9.30) is seen to hold, i.e., (9.27) is the negative of the transpose of (9.26).] In combining (9.30) with (9.18), (9.18'), and (9.21), it will be seen that, if any one of RpK7D,",Mpa, and N," alone is given, all the others can readily be obtained by merely permuting columns, transposing, changing the sign, etc. Theorem 9.5 can be further strengthened to yield

Theorem 9.8. If ( s23) is loop-free and 23,( & 23) is cutset-free, and, then there exists a treein addition, if they are disjoint, B1n 23, = 0, cotree pair (2,2) such that 2 2 B1and 2 2 23, .

PROOF. If 23, u 23, = 23, then 23, must be a tree itself, and 23, a cotree itself. For the number of branches of B1 does not exceed m,that of 23, does not exceed k , and their sum is equal to n = rn + k, so that 23, has exactly m branches and 23, exactly k , and consequently they are a tree and a cotree. Otherwise, choose a branch b, E 23 - (B1u !B2).Then, either 23, u {b,} is loop-free or 23, u {b,} is cutset-free, because, if we assume that there exist at the same time a loop

6=

t b K

+

c

<"bl

(5

?PbP

(? Z 0 )

bAE'81

and a cutset

+

c

b p ~ ' 8 2

such that b, E car 6 &

'B1u {b,}

and

b, E car q E 23, u {bK},

B1n Bj2= 0will lead us to the contradiction o = t . q = 5qZO. Let 23,' = B1u {b,}, 23,' = 23, if blu {b,} is loop-free, or 23,' the condition

=

54

111. Topological Properties of Networks

232' = Bzu {bK} otherwise. Then Bl' is loop-free, B2'is cutset-free, and B l ' n 23z' = 0. Continuing in this way, we can gradually augment the

mutually disjoint loop-free subset and cutset-free subset until we finally get to a tree-cotree pair. Q.E.D. 9.4. Bases for Loops and Cutsets

{cl,

ck}

When . . . , is a maximal independent set of loops, we call the matrix RPKin the expression (9.3 1) a loop matrix. When { q ' , . . . , q'"} is a maximal independent set of cutsets, we call the matrix D,"in the expression n

1 D,"b,

q"=

( a = I , ... , n ~ )

(9.32)

K = l

a cutset matrix. A maximal independent set of loops

{gl, . . . , kk) is called a basis of X(G), or a basisfor iiitegrul loops, if every loop in X(G) is expressible as a linear combination of 5;s with integer coefficients (in a unique way). Similarly, we define a basis of Y(G), or a busis fur integral cutsets, {q', . . . , q"'}. Obviously, the primitive set of loops or cutsets introduced in 99.3 is a basis for integral loops or cutsets. It should bc noted, however, that there exists a basis for loops as well as that for cutsets which is not a primitive set associated with a tree-cotree pair. For example, in Fig. 9.3(a), 51

= b, - b2,

52

= b2

- b3,

53

= b3

- b,

is a nonprimitive basis for loops, and, in Fig. 9.3(b), 91 = b, - b,,

9 2

= b2 - b3,

q 3

= b3 - b4

is a nonprimitive basis for cutsets. (The reader should verify that there is no tree-cotxee pair that has either of the preceding basesas the primitive set.) It is also worthwhile to note that a maximal independent set is not necessarily a basis. For example, in the graph of Fig. 9.4, the loops -b2 +b, 52

=

63 =

b, -b1 +b2

+b5 +b,,

-b3 +b4

+ b4 + b5

+b,,

9. Loops, Cutsets, Trees, and Cotrees

(a)

55

(b)

Fig. 9.3. Nonprirnitive integral bases.

form a maximal independent set. But, the fractional coefficients are needed in order to express the loop

6 = b1 - b,

+ b,

as their linear combination :

5 = tS1 + f S 2 - $ 5 3 . In contrast with this, when we consider V(R, b), V(R, a),X(R), Y(R), etc., with the "field" of rational or real numbers as the domain of coefficients, a maximal independent set of loops or cutsets is always a basis (cf. Remark 9.2). REMARK 9.1. It is sometimes expedient to make use of those qa's which are defined by n

q"=

C

D,"b,

( ~ = l..., , m),

(9.33)

lC=l

where B,"

the basis

56

111. Topological Properties of Networks

of Y(G). That these q a ' s form a basis can be shown as follows. An arbitrary cutset q E Y(G) is written as 9 =4 M

with a suitable

5 = C lana E V(G, a),i.e., a= 1

(9.34) Substitution of (8.15) in (9.34) gives

M

where y ,

=

C A,"[, a=

is an integer for every a. The uniqueness of the

1

expression (9.35) follows from the fact that that the qa's are mutually independent.

D," is

of full rank rn, i.e.,

REMARK 9.2. Let us give a few comments on the case where we adopt the field R of rational or real numbers, instead of the ring G of integers, as the domain of coefficients of vector spaces. V(R, b), V(R, %), X(R), Y(R), etc., are defined formally in the same manner as V(G, b),V(G, %), X(G), Y(G), etc., respectively, with R in place of G wherever it appears. As modules, the former modules contain the latter as submodules. The various properties related to loops and cutsets hold mutatis mutandis. Especially, it remains to be the case that the primitive sets of loops and those of cutsets, as well as the set mentioned in Remark 9. I , are the bases for X(R) and Y(R). However, the distinction no longer remains between the bases and the maximal independent sets of loops and cutsets. As will become evident from the expression of a loop or a cutset in the form of a linear combination of the primitive ones associated with a tree-cotree pair, we have Theorem 9.9. A loop

4=

n

t"b,

[E

X(G)] is uniquely determined by

,= 1

those 5,'s (arbitrarily given) which correspond to the branches of a cotree. n

A cutset q

=

C v], K=l

b,

[E

Y(G)] is uniquely determined by those qK's

(arbitrarily given) which correspond to the branches of a tree.

57

9. Loops, Cutsets, Trees,and Cotrees

Next, we shall investigate the relations between one basis and another.

Let {rp(p = 1, . ..,k ) } be a basis for X(G) and {rp'( p = 1, .. ., k ) } another. Since every rp' is an element of X(G), it is represented as a linear combination of r i s : k

c

( p = 1,. .. , k ) ,

rd = q = 1 A,%,

(9.36)

where the A , ~ ' sare integers, and, similarly, the converse is also true: k

c

( p = 1, ..., k ) ,

r p = q = l B,4r,'

(9.37)

where the BPq'sare integers. Since k

rs

s= 1

( p = l , . . ., k)

(9.38)

and since the rP)s are mutually independent, we must have (9.39) Hence, denoting the determinant of a matrix, say Apq,by det(Apq), we have 1 = det(6,")

= det(Bpq)det(Ad),

(9.40)

so that both det(B,4) and det(A,"), being integers, should be equal to 1 or - 1, at the same time. A similar method is used for the bases for cutsets. The relation such as (9.36) is rewritten in terms of loop matrices and cutset matrices to yield the following theorem. Theorem 9.10. For two cutset matrices D," (a = I , . . ., r n ; K = I , and 0:"(a = 1,. . ., r n ; K = 1, .. ., n) corresponding to different bases, there is a transformation matrix A,," with integer elements whose determinant is 1 or - 1 (and, consequently, whose inverse also consists of integer elements alone), such that

. . ., n)

m

D;=

C

Ab"D,b.

(9.41)

b= 1

The same process may be applied to loop matrices RpK( p = 1, . . . ,k ; K = 1, . . ., n) and R r ( p = 1, .. ., k ; K = 1, . . ., n) corresponding to different bases.

58

111. Topological Properties of Networks

Let us consider in more detail a particular case of basis transformations, i.e., transformations among primitive bases. Let us take a tree-cotree pair (2,%) and another pair (2', 2') such that 2 and 2' (and, consequently Z and T') differ from each other by only one branch, i.e., let us assume that the aoth branch of 2 (which we denote by b,) is the poth branch of 2'and the poth branch of 2 (which we denote by b,J is the aoth branch of 2',all the other branches being common for 2 and Z'as well as for % and 2'. If D," (with row index a and column index K) is the primitive cutset matrix associated with (2, T) and 0:"that associated with (2', Z'), then, as is seen without difficulty from the special form (9.21) of a primitive cutset matrix, the matrix 0:" is obtainable from D," by the elementary row operation (or the sweeping-out operation) which brings the KZth column of D," to the unit vector having 1 in the aoth row. (Incidentally, by this e!ementary row operation the rc,th column of D," is changed from a unit vector to a nonunit vector.) Similarly, the primitive loop matrix R; (with row index p and column index K)associated with (2', T) is obtained from the primitive loop matrix RPKassociated with (2, 2)by the elementary row operation which brings the K~ th column of RPKto the unit vector having I in the path row. (The reader should clarify how the matrices M," and N,' are transformed by the basis transformation of this kind.) The case where Z and 2'(or T and Z') have more than one noncommon branch can be reduced to the foregoing case as follows. If 2 # 2' (and hence T # T'), then there is a branch bKZwhich belongs to 2'but not to 2. Sincc Zu{b,,} is not loop-free, there is a (minimal) loop 6 such that car 6 s 2 u (bkz). (Note that car 5 is uniquely determined.) Since 2' is loop-free, there should exist a branch bKl which belongs to car 6 but not to 2'(b,, should belong to 2 because b, E 2').Then 2 = Z u { b K 2 } {b,,} is loop-free and has the same number of branches as 'I,so that it is a tree. 'Iand T,have only one noncommon branch on the one hand, and, on the othcr, the number of branches common to Z,and 2' is greater by one than thc number of branches common to 2 and 2'.In this manner, for arbitrary two tree-cotree pairs (X, 2)and (2') Z'), we can find pairs -

(2, T) = (20,To), ( T I ,Q, . . . , (21-1, ZI-I), (21,TI) = (Z'2') , such that

and Z,(or Z , - I and X I ) have all but one branch i n common.

9.5. Transformations of Graphs and Matrices I t is both of theoretical significance and of practical use to investigate to what kinds of transformations loops, cutsets, trees, cotrees, incidence matrices, loop matrices, cutset matrices, etc., are subject when the under-

59

9. Loops, Cutsets, Trees, and Cotrees

lying graph is transformed by open-circuiting and/or short-circuiting branches (cf. $7.1). The statements to be described in the following are intuitively so evident that we may not write out the formal proofs, leaving them as the exercises for the reader. Let us fix a subset Bo= {bKl,. . . , bKr}(sB)of branches of a graph 6, and consider the graph 6, obtained from 8 by short-circuiting all the branches of B0. The set of branches of 6,, therefore, coincides with (or, more precisely, is put in one-to-one correspondence with) the set B, = 23 - bo.Our problem is to investigate the relations between loops, cutsets, etc., in Q and those in 8 , . To every loop 4, = ("b, in V(G, 8,) of GI we can find a corresponding loop

6=

c 5, + c tab4in V(G, 23) of 8 (not necessarily unique) bK

b iE s o

in the following manner. 5 , is a linear combination of elementary loops in V(G, Sl), to which there are the corresponding closed paths in GI. Each of these closed paths can be augmented by some branches of Boto yield a closed path in 6,and we may take as 6 the linear combination (with the same coefficients) of the elementary loops in V(G, 23) corresponding to such closed paths in 8. Especially, if !B0is loop-free, 6 is determined uniquely by 5,. Conversely, to every loop 6 = ("b, t A b Ain V(G, 23) of 8 we

+2

bKE8i

bzEso

can make correspond in the unique way the vector

6 , = t"b, in of 6 to V(G, 23,) bKEsi

V(G, 23,)of 6,which is a loop, 5, being the projection of (GI. A cutset q , = C q K b, in V(G, 23,) of 8 , may be regarded as a vector b,ESi

in V(G, 23) of 6 which is a cutset. In fact, each segregation in 6, corresponding to one of the elementary cutsets, of which q, is expressed as a linear combination, is a segregation also in (5. Conversely, every cutset q in V(G, 23) of (6 whose carrier is included in 23, is a cutset also in V(G, 23,) of (5,. [A cutset q in V(G, 23) of 8 such that car q n do# 0 does not in general yield a cutset in V(G, 23,) of through projection.] The foregoing relations are summarized in the following theorems, which may be verified directly. (Theorem 9.11 follows directly from the definition of the operation of short-circuiting; cf. $7.1 .)

Theorem 9.11. Let 6," be the incidence matrix of 8 (with row index ci and column index K.) Then the incidence matrix of the graph obtained from 8 by short-circuiting a branch bKlcan be constructed from 6," by replacing the two rows corresponding to a+ bKl and 8-bKlby one row that

60

Ill. Topological Properties of Networks

is equal to their sum and then deleting column K ~ The . incidence matrix of 6,is obtained by applying successively the similar operations for b,, * *, Theorem 9.12. The necessary and sufficient condition for a subset 2 of branches including 8,to be a tree on 6 is that 8,is loop-free and 2 - 2&, (g8,) is a tree on GI. Theorem 9.13. Let 2 ( 28,)be a tree on 6,Z its complement cotree, and D," (with row index a and column index K) be the primitive cutset Then the primitive cutset matrix of matrix of 6 associated with (2,Z). 6,associated with (2- B,, Z)-note that B1- (2- 8,)= T-is obtained from D," by deleting the 1 columns and the I rows corresponding to the branches b K l , .. ., bKl of 23,. The primitive loop matrix of 8, associated with (2- So,2) is obtained from that of 6 associated with (2, T) by deleting the I columns corresponding to the branches of 8,. The matrix M,," or NPa[introduced in (9.21) and (9.lS')l of Q1 associated by with (2- 8,, 2) is obtained from that of 6 associated with (2,Z) deleting the I columns or the 1 rows corresponding to the I branches of 23,.

Next, we shall consider the graph 6, obtained from 8 by opencircuiting all the branches of 23,. This time, we shall denote the branch set of 6,by 8, = 8 - 8,. Every loop 6, in V(G, 8,)of 6, can be regarded as a vector in V(G, 8) of 6 which is also a loop. Conversely, every loop 6 in V(G, 8)of 8 such that car 6 g 8,can be regarded as a loop in V(G, 8,)of 6,. To every cutset q, = C q, b, in V(G, 8,)of 6,we can find a correbc€232

sponding cutset q

= q,

+ C

qAb, in V(G, 23)of 6 (not necessarily deter-

bis23o

mined uniquely) in the following way. q, is a linear combination of elementary cutsets in V(G, B,), to which there are the corresponding segregations in 6,.Each of these segregations can be augmented by some branches of 23, to yield a segregation in 6, and we may take as q the linear combination (with the same coefficients) of the elementary cutsets in V(G, 8)corresponding to such segregations in 6.In particular, if 8, is cutset-free, q is uniquely determined by q2. qAb, in V(C, 8)of 6 Conversely, to every cutset q = qK b,

1

b,

E 232

+

bnsBo

q, b,

we can make correspond in the unique way the vector q,= blcE23z

the projection of q to V(G, 8,) of 6,,which is a cutset in 8,.

9. Loops, Cutsets, Trees, and Cotrees

61

Theorem9.24. The incidence matrix 6: (with row index LX and column index ti) of 6, is obtained from that of 6 by deleting the I columns corresponding to the branches bKl, . .., b, of 23,. Theorem 9.15. The necessary and sufficient condition for a subset Z of branches including 23, to be a cotree on 6 is that 23, is cutset-free and T - 23, (g23,) is a cotree on 6,. Theorem 9.26. Let 2 ( 223,) be a cotree on 6, 2 its complement tree, and RpK(with row index p and column index ti) be the primitive loop matrix of Q associated with (2,%). Then the primitive loop matrix of 0, associated with (2, Z - 23,)-note that 23, - (T - 23,) = %-is obtained from RpKby deleting the I columns and the I rows correspotiding to the I branches of 23,. The primitive cutset matrix of 6,associated with (2,Z - 23,) is obtained from that of 6 associated with (2,%) by deleting the I columns corresponding to the I branches of 23,. The matrix Mp" or N," [introduced in (9.21) and (9.18')] of 8 , associated with (2,% - 23,) is obtained from that of 6 associated with (2,T) by deleting the I rows or the I columns corresponding to the I branches of 23,. 9.6. Unimodularness of Primitive Loop Matrices and Primitive

Cutset Matrices Let us fix on a graph 6 a tree-cotree pair (2, 2)and, associated with it, the primitive basis of loops {r,, ( p = 1, . . ., k ) } , the primitive basis of cutsets {aa (a = 1, . . . , rn)}, the primitive loop matrix RpK,and the primitive cutset matrix D," (K = 1, ...,n). We shall denote a subdeterminant of the matrix of D," with row indices [a,, . . . , a,] and column indices [ K ~ .,. . , K J (where we should distinguish only between the even and the odd permutations of the same set of indices) by

D;:;.::;,.

(9.42)

We shall use similar notations for subdeterminants of RpK.Then we have the following theorem. Theorem 9.27. An rn x rn subdeterminant Dki;::ZLl of a primitive cutset matrix D," (a = 1, .. . ,m ; K = 1, ..., n) is equal to 1 or - 1 if branches bKl,. . ., bKmform a tree and is equal to 0 if they do not form a tree. A k x k subdeterminant RE;!::$k1of a primitive loop matrix R,K ( p = 1, . .., k ; K = I , . . . ,n) is equal to 1 or - 1 if branches b,,, . . . , bKk form a cotree and is equal to 0 if they do not form a cotree. (Here a tree or cotree may or may not be that with which the D," or RpKis associated.)

62

111. Topological Properties of Networks

PROOF. We shall give a proof for D," only. The statement for RPK will be proved similarly. Let the associated tree of D," be 2. If {bKl, . . . , b,,} = 2' is not a tree, it is not loop-free (Corollary 9.l), so that there is a 171

nonnull loop

6=

(",bKx,which is orthogonal to all the cutsets of the i= 1

primitive set {d', . , . , d"}. Therefore, we have

-

0 = 5 d" =

(

i= I

<",b,,)

.

( f D."b,) K=

m

=

C DtitKi i=

(a = 1 , .

.. , m )

1

1

(9.43) with at least one nonnull tKi.Hence D[l...ml [KI"'K,]

=

Next, let us assume that {b,,, . . ., }b,, = 2' is a tree. If we denote the primitive cutset matrix associated with 2' by D: (a = I , . ..,m ; i i = I , . . . , n ) , we have a matrix A," with the determinant of absolute value 1 such that ...

( a = 1,..., m ;

A,"D,b

0:" =

K =

1 ,..., n),

(9.44)

b= 1

by Theorem 9.10. Hence,

where det(A,")

=

f 1. Obviously, D'"K i

=0

( a # i)

=1

( a = i),

(9.46)

so that DI [ 1 ...m

] [K!...K,,,]

=

(9.47)

Therefore,

Theorem 9.17, when combined with the descriptions in 59.5,will lead to the following interesting property of the primitive matrices.

Theorem 9.18. Every primitive loop matrix, as well as every primitive cutset matrix, is totally unimodular, where a totuNy unimodulur matrix means a matrix each of whose subdeterminants is equal to 1, - 1, or 0.

9. Loops, Cutsets, Trees, and Cotrees

63

PROOF. As before, we shall give a proof for primitive cutset matrices only. Let 2 = {bdl, . .., barn}be a tree, {d', . . . , dm}be the associated primitive set of cutsets, D," (a = 1 , . .., m ; K = 1 , . .., n) be the associated be its subdeterminant of order primitive cutset matrix, and DE;;:::;:; I (Sm)with rows a,, ...,a1 and columns K ~ ,. . . ,K ~ We . may assume that 1 < m,since the case I = m was already dealt with in Theorem 9.17. Let { u ~ + .~.., , a,} = (1, .. ., m } - {al, . . ., a,}, and let {bnl+l,. . ., bAm}be the branches of 2 corresponding to d""', . . . , dam. If {b,l+l, . . . , bAm}n {bK1,. . . , bKl} is nonempty, i.e., if it contains a branch, say bl,o = bKJo ( I + 1 i, 5 rn, 1 5 j , 5 I), then bKJo,belonging to 2, appears in none of d', . . ., d" except in d"'0, so that DOi

KJo

=O

for i = 1, ..., 1.

(9.48)

Therefore, in this case, we have In the remaining case, where { bAl+ . . . , barn)is disjoint from { bKl, .. . , bK1}, we consider the graph B2that is obtained from the original graph 6 by short-circuiting the branches b, . . ., barn. Since the short-circuited branches form a loop-free subset in 6,{bd1, . . ., b,} is a tree on Q2 (Theorem 9.12), and the primitive cutset matrix of Q2 associated with this tree is obtained from D," by deleting rows a[+,, .. . , a, and columns A,,,, . .., I,, (Theorem 9.13). Therefore, in this case, D;;;;::::l may be regarded as a subdeterminant of the primitive cutset matrix of Since, furthermore, the rank of Q 2 is obviously equal to I, we can apply Theorem 9.17 to get the conclusion that Or[;;:::::: is equal to either +1 or 0. The proof of the unimodularness of primitive loop matrices is done similarly (cf. $10). Q.E.D. REMARK 9.3. b/ (incidence matrix), 6,"(reduced incidence matrix), M:, and N," are also totally unimodular. The proof is left to the reader. (Hint: the total unimodularness of M,," or N," can be reduced to that of D," or R,K. For 6," and D,",first establish the proposition corresponding to Theorem 9.17, and then consider the graph obtained by open-circuiting all the branches other than b,, . .., b,,.) +

9.7. Series and Parallel Branches Two branches b, and b, (K # 1) (of which neither forms a segregation by itself) are called series if there is a cutset q such that car q = {b,, b,} [see Fig. 9.5(a)]. If b, and b, are series,.then the aforementioned cutset is

64

111. Topological Properties of Networks

determined to within a constant factor, and if b, and b, are series and b, and b, are series, then b, and b, are series. (If ql = q, b, q, b, and q2 = qn’bA q, b, are cutsets with q , , q,, q,’, q, # 0, then q3 = qn’ql - q, q2 = q , ’ ~ b ,, - q, q, b, is a cutset, and car q3 = { b, , b,}.) More than two branches are called series if any two of them are series. Two branches b, and b, (K # A) (of which neither is a self-loop) are called parallel if there is a loop 6 such that car 6 = { b, , b}, [cf. Fig. 9.5(b)]. If b, and b, are parallel, then the aforementioned loop is determined to within a constant factor, and if b, and b, are parallel and b, and b, are parallel, then b, and b, are parallel. More than two branches are called parallel if any two of them are parallel.

+

+

(a)

(b)

Fig. 9.5. (a) Series branches; (b) parallel branches.

The intuitive meanings of series and parallel branches will be almost evident from Fig. 9.5. The following theorem gives another characterization of the concept of series or parallel. Theorem 9.19. Branches b, and b, are series if and only if there is a loop whose carrier contains either b, or b, and, in an arbitrary loop n

5=

tpb,, ,= 1

the coefficient 5” for b, is equal to that

tAfor b,

except for a

sign, i.e., lyl = 15’1. Similarly, branches b, and b, are parallel if and only if there is a cutset whose carrier contains either b, or b, and, in an arbi-

PROOF.If b, and b, are series, there is, by definition, a cutset q whose carrier they contain. Since neither b, nor b, is a cutset by itself, q is minimal

65

9. Loops, Cutsets, Trees, and Cotrees

and hence a multiple of an elementary cutset (Theorem 9.2). We therefore assume that = b, - Eb, (9.49) where

E

is 1 or - 1. Let

6=

c rpbp be a loop. Then it follows from n

p= I

Theorem 9.4 that

0 = 5 * q = r " - €5'

or

5" = E ~ I ,

(9.50)

i.e.,

lt"l = 15".

(9.51)

Since b, is not a cutset, there is a loop 5 such that 5 * b, # 0. Obviously, car 6 3 b,. Conversely, let us assume that, in every loop 6, (9.51) holds. Then (9.50) holds for every loop 5 with one and the same E (= 1 or - 1). For, otherwise, there would be loops

61 =

c c

5Ipbp

+

51bK

- 5 1 b,

(51

# O),

529I

+

52

b,

+ 5 2 b,

(52

# 0);

(9.52)

P+K,I

52

=

pf~,A

however, we should have 5 2 51

1 + 5 1 5 2 = P+,,,

( 5 2 51"

+ 51 5 f P p + 25, t z b, + 0 . b,,

a loop in which the coefficient of b, is 25,5,, that is nonnull, and that of b, is 0, which contradicts the assumption. Therefore, the vector q = b, - Eb, is orthogonal to every loop so that it is a cutset (Theorem 9.4). If b, itself is a cutset, it is orthogonal to every loop so that no loop can have the carrier containing b,. Hence, b, is not a cutset by itself. Since q is a cutset, b, is not a cutset, either. The proof for the case of parallels Q.E.D. can be done in much the same way. Since the concepts of series and parallel branches are defined in terms of loops and cutsets, they can be characterized in terms also of a primitive loop mauix RpK,a primitive cutset matrix D,", matrix MP",or Npa.Furthermore, as we have seen in $9.3, any statement in terms of one of these four kinds of matrices can directly be restated in any other, so that we shall show the characterization of series and parallel branches in terms of the matrix N,' (with row index a and column index p ) associated with a treecotree pair (2,2).As was pointed out, the rows of N,' correspond to the branches of the tree 2 and the columns to the branches of the cotree 2.

66

111. Topological Properties of Networks

We first observe that no two branches of 2 can be in parallel (since 2 is loop-free) and that no branches of T can be in series (since ?Z is cutset free). For two branches of 2 to be in series, it is necessary and sufficient that the sum or the difference of the corresponding rows of N,” should vanish, i.e., that those rows coincide with each other to within a sign. For the branch of 2 corresponding to the ath row of N,“ to be in series with the branch of Z corresponding to thepth column of N,”, it is necessary and sufficient that the ath row of Np“ has 1 or - 1 in the pth column and all the other elements of that row vanish. Two branches of Z are in parallel if and only if the corresponding columns of Npa coincide with each other to within a sign. The branch of Z corresponding to the pth column of Npa is in parallel with the branch of 2 corresponding to the ath row if and only if the pth column of N,“ has 1 or - 1 in the ath row and all the other elements of that column vanish. 10. DUALITY

As has perhaps been noticed by the reader, all the concepts, as well as their properties and relations, expounded in the foregoing sections concerning a graph, that are stated in reference to “branches” (their subsets and the vector spaces on them) but not to “nodes” have a “dual” structure as a whole, in the following sense. Let us take up an arbitrary definition or theorem that has no reference to nodes. If we perform the simultaneous replacement of terms such as branch cf branch rank ++ nullity loop cf cutset tree t+cotree series tt parallel short-circuit tt open-circuit closed path cf segregation (as a set of branches) (as a set of branches)

in that definition or theorem, then we have a sentence that coincides in meaning with a definition or a theorem described elsewhere. The sentences related to each other in this way are said to be dual to each other, and the structure of this kind of the theory is called the duality. Thus, Theorems 9.1 and 9.2 are dual to each other; Theorem 9.3 is dual to itself, i.e., self-dual; Theorem 9.4 is self-dual; the definition of a tree is dual to that of a cotree; Theorem 9.5 is self-dual; the definition of a primitive set of loops, a primitive loop matrix, or a matrix M,’ is dual to that of a primitive set of

67

11. %Isomorphism

cutsets, a primitive cutset matrix, or a matrix NPa,respectively; Theorem 9.6, as well as Theorem 9.7, is self-dual; Theorem 9.8 is also self-dual; the definitions of bases, as well as Theorems 9.9 and 9.10, in 99.4, are self-dual as a whole; similarly, it will be seen that the descriptions in 99.5, as well as those in $9.6, have the dual structure; the unusual definitions of series and parallel branches as given in 99.7 are adopted for the purpose of realizing the dual characterizations of them. It is obvious that the whole theory that will be developed on the basis of the concepts and their relations that are endowed with a dual structure will have also a dual structure. In the following sections and chapters, we shall try to develop our expositions in the dual form as far as possible. We may develop a theory of graphs and networks in a complete dual form starting from a system of axioms which is self-dual. This would be theoretically more elegant than the manner in which we have presented this book (see, e.g., Whitney [4] and Minty [2]). However, it would be necessary to prove, in some way or other, that the intuitive graphical images of the primitive concepts appearing in the axioms satisfy all the relations stated in the axioms. A less elegant way would be to adopt all the descriptions in the foregoing sections which can be put in the frame of dual structure as the system of axioms and to base the subsequent development of the theory upon them. Although they will not be independent, their consistency is guaranteed by the existence of a model, i.e., a graph. However, we shall not do so, because a theory that is completely self-dual cannot deal with the nondualizable concepts such as “nodes” within its framework. Although very infrequently, it happens that ‘‘nodes ” are of essential use for establishing practically efficient algorithms. We do not want to sacrifice the practical significance to the elegance and symmetry of the theory. A compromise is to intend to put the descriptions in the framework of dual structure whenever it is possible, and we shall try to do it throughout this book. The reader is asked to go through the book examining which part of the theory and the algorithms has the dual and which part has not. 11. 2-ISOMORPHISM

11.1. Articulation Points and Two-Terminal Subgraphs Let us dissect the set of branches b of a graph 8 into two subsets Bl and b2(Le., b = blu b2 and B, n b2= and let us denote the set of those nodes of 8 which are incident to the branches of B1by ‘illl and

a),

68

HI. Topological Properties of Networks

the set of those nodes which are incident to the branches of 23, by a,. [Note that we have 9Il n %, # in general, and that % - (%, u a,) consists of isolated nodes.] If there is a dissection (B,,23,) of 23 such that n %, consists of a single node, we call the node an articulation point. The node incident to a self-loop is always an articulation point unless the self-loop constitutes a connected component by itself. In regard to an articulation point of a graph 8,we have the two subgraphs 8, and O2 of 8,each of which consists of and bi ( i = 1 or 2), and we shall say consisting of the connected components each of which that the graph 8’, is either a connected component of 8,or one of 8,,is obtained from 8 by cutting at the articulation point. In a similar fashion, if %, n %, consists of exactly two nodes, we call the subgraph 8,of 8 constituted of %I and 8,a two-terminal subgraph. The complement of 8,in regard to 8,i.e., the subgraph 0, constituted of %, and 23, , is also a two-terminal subgraph. This definition of two-terminal subgraphs, however, allows a single branch, a parallel connection of branches, or its complement to be called a two-terminal subgraph, which is too trivial to call such. Therefore, we sometimes exclude such a trivial subgraph from among the two-terminal subgraphs; if 0 is connected and has no articulation point, this is equivalent to assume that a two-terminal subgraph should have more than 2 and fewer than M nodes (where M is the number of nodes of the entire graph).

EXAMPLE.The graph of Fig. 7.3 has two articulation points n3 and n 4 . It has, e.g., a two-terminal subgraph consisting of nodes {nl, n 3 , n4, n5} and branches {b, , b, , b, , b, , b9}. We say that a graph 8‘is obtained from 8 by reversing a two-terminal subgraph of 8,if 8’has a two-terminal subgraph Q1’ with the complement 8,’such that: is isomorphic with the complement 8,of 8 , (in 8), (i) 8,’ (ii) 8,’ is isomorphic with the graph obtained from 8, by reorienting all the branches, and (iii) if 8, and 0,have nodes nu and nB in common, then 6,’and 8,’ have two nodes nor’and n8‘ in common, where na‘ corresponds to na and n8’ to n8 under the correspondence (i) between 8,’and 8,, and, under nu’ corresponds to nb and nB’ correspondence (ii) between GI’ and 8,, to n.. The intuitive meaning of the reversion operation is to cut off a twoterminal subgraph from its complement at the two common nodes, to

69

11. 2-Isomorphism

reorient all the branches of the former subgraph, and then to reconnect them in such a manner that they may have the same common nodes but that the manner of connection may be different (cf. Fig. 11.1).

(62

(6,

(Gr‘ = 6j2

(6,’

(6 ’

6

Fig. 11.1. 2-Isomorphic graphs.

11.2. Separability

A matrix is said to be separable if it is brought into the form shown in Fig. 11.2 by appropriate permutation of columns and of rows. If a primitive cutset or loop matrix of a graph is separable, then so is every other primitive cutset or loop matrix. The proof of this fact is easily found by considering the manner of transformation among primitive matrices associated with different tree-cotree pairs (cf. the arguments toward the end of $9.4 as well as the relation between the primitive loop matrix and the primitive cutset matrix associated with the same treecotree pair). (The reader should try to find the expression of the same thing in terms of matrices Mpaand N,”.) A graph is said to be separable if one of its primitive cutset and loop matrices is separable. A graph that is not

Fig. 11.2. Separable matrix.

70

111. Topological Properties of Networks

separable is said to be nonsepurable. A maximal nonseparable subgraph of a graph 6 is called a nonseparable component of 8. As is easily seen, the set of branches 23 of (I, is decomposed uniquely into the subsets of nonseparable components, and the decomposition into the nonseparable components is a refinement of that into the connected components (cf. $8). We can decompose a graph into the nonseparable components by cutting its connected components at the articulation points. A graph is nonseparable if and only if it is connected and has no articulation points. (The reader should prove these propositions.) EXAMPLE.The graph o f Fig. 7.3 is separable. It has four nonseparable components {b,, b 2 , b 3 , b 5 , b,, b,, b d , {b4), { b d , and PI,, bid.

11.3. 2-Isomorphic Graphs Two graphs (I,, and 6, are said to be 2-isomorplzic if there is a one-toone correspondence between the set of branches B1of 6, and that of 23, of 8,and, under this correspondence, every loop in corresponds to a loop i n (c2, and vice versa. It is evident that 2-isomorphism is an equivalence relation. We can restate the definition of 2-isomorphism in many different ways. For example, we have an equivalent definition by replacing ‘‘loop’’ in the preceding definition by “cutset” (Theorem 9.4). Or, if the branches of (I,, corresponding to a tree on 6, also form a tree and the primitive cutset matrix of 6,and that of 6, associated with the respective trees are equal to each other, then O1 and 8,are 2-isomorphic, and vice versa. The same process holds for primitive loop matrices, matrices Mp”’s, or Npe’s. If 6,and E2are 2-isomorphic, a tree or cotree on one corresponds to a tree or cotree on the other, respectively. The converse is not true. However, if every tree on 6, corresponds to a tree on 6,, and vice versa, then 8, and 6, are 2-isomorphic to within reorientations of branches. We say that a property in regard to graphs is invariant in regard to 2-isomorpIiism, if, whenever it is possessed by a graph, it is possessed by any other 2-isomorphic graph. Thus, the separability of a graph is invariant with regard to 2-isomorphism, but the connectedness is not. The following theorem, which was proved by Whitney [3], clarifies the intuitive significance of 2-isomorphism in terms of the concepts introduced in $1 1.1. The sufficiency part of the theorem is obvious, whereas the necessity part requires lengthy (though elementary) proof, so that we shall omit the proof.

71

11. 2-Isomorphism

Theorem 22.2. For two graphs to be mutually 2-isomorphic it is necessary and sufficient that we can transform one of them into the other using the transformations of the following five kinds in an appropriate order:

(a) To reorient all the branches belonging to a nonseparable component. (b) To reverse a two-terminal subgraph (cf. $11.1). (c) To combine two connected components into one by identifying a node of one of them with a node of the other. (d) To cut a (separable) connected component at an articulation point. (e) To add or remove isolated nodes.

REMARK 11.1. Among the five kinds of transformations in Theorem 11.1 which produce 2-isomorphic graphs, only (b) is nontrivial. A transformation of the kind (c) decreases the number of connected components by one. The transformations of the kind (d) are inverse of those of the kind (c).

EXAMPLE 1. The graph of Fig. 11.3 is 2-isomorphic with that of Fig. 7.3. The former is obtained from the latter by cutting at the articulation point n3 , reversing the two-terminal subgraph containing branches {b, , b, , b9}, combining the connected component {blo, b, ,} with the largest connected component, cutting off the nonseparable component {b,} and combining it at another place, and then removing the isolated node n,.

b,,

Fig.11.3. A graph that is 2-isomorphic with the graph of Fig. 7.3.

72

111. Topological Properties of Networks

(b)

(a)

Fig. 11.4. Fan of triangles and triangular bridge-frame.

EXAMPLE 2. The graph of Fig. 11.4(a) is 2-isomorphic with that of Fig. 11.4(b). In fact, in either of them, branches {bl, b, b,, b, b,, bs} form a tree, and the primitive set of loops associated with it is {b, + b, b, b, - b, - b, , b, b, b, b, - b, - b,, b, b, b6}. Figure 11.4(b) is obtained from Fig. 11.4(a), as shown in Fig. 11.5, by reorienting successively the two-terminal subgraphs enclosed by broken lines. It should be noted that the 2-isomorphism of this kind, i.e., between a fan of triangles and a triangular bridge-frame, exists for any number of triangles.

+

+ +

+ +

Fig. 11.5. 2-Isomorphic transformation of a fan of triangles into a triangular bridge-frame.

12. Planar Graph and Its Dual

73

12. PLANAR GRAPH AND ITS DUAL

If a one-to-one correspondence is defined between the branch set 23, of a graph 8,and the b2 of another graph 8,and if, under this correspondence, every loop in 8,corresponds to a cutset in 8, and every cutset in 8,to a loop in 8,(or, equivalently, if every cutset in 8, corresponds to a loop in Q2 and every loop in 8,to a cutset in 8,;cf. Theorem 9.4), then 8,is called a dual of 8 , .From this definition, it directly follows It is also obvious that, that, if, is a dual of 8,, then 8,is a dual of 8,. if 0,and G2 are dual of each other, any graph 2-isomorphic with 8,and any graph 2-isomorphic with Q2 are dual of each other, and that, if G2 is a dual of 8,and O3is a dual of @ j j 2 then , 8, and (lj3 are mutually 2isomorphic. Every tree on a graph corresponds to a cotree on its dual, and vice versa. Not all the graphs have dual. A graph is called a planar graph if it has a dual. A dual of a planar graph is, therefore, also planar. The term " planar " stems from the following theorem. Theorem 12.1. A necessary and sufficient condition for a graph to be planar is that it can be drawn on the (two-dimensional) Euclidean plane (or on the two-dimensional sphere) with no crossing of branches. As we shall do just below, it is easy to see that the condition is sufficient, but, as for its necessity, we must beg the reader to refer to the original paper (see Whitney [l]). We can construct a dual G2 of a graph Q1 drawn on the Euclidean plane in the following way. Since 8,is drawn on the plane, its branches partition the entire plane into M 2 = n, - rn, 1 = k, 1 regions (of which one is unbounded and the rest are bounded), where n, is the number of branches of 8,, and rn, and k, are the rank and the nullity of 8,. We define the set of nodes f12 = {n;, ..., rill;,} of 82 by putting an n,* in each one of those regions, and the set of branches Bj2= {b?, . . . , b:*} (n, = n,) by connecting nx and n; by b,* if and only if the branch b, of 8,bounds the regions in which n,* and n$ are put, where the orientation of b: is determined in such a way that, if we regard the corresponding branch b, of 8, as the river flowing in the direction of its arrow, the b: is directed from the right bank to the left bank. Under the obvious oneto-one correspondence between 23, and d 2 ,it follows immediately from the manner of constructing O2 that each elementary closed path in 8, corresponds to an elementary segregation in 6, and each elementary Therefore, G2 is a segregation in G2 to an elementary closed path in 8,. dual of 8,(cf. Theorems 9.1 and 9.2).

+

+

74

111. Topological Properties of Networks

In concluding, we mention, without proof, the famous theorem originally presented by C . Kuratowski (see Kuratowski [l] and also Whitney [2]) in our own terminology. Theorem 12.2. A necessary and sufficient condition for a graph to be planar is that neither complete pentagon nor Thomsen graph arises from it by short- and open-circuiting branches (see Fig. 12.1).

REMARK 12.1. The two graphs in Fig. 12.1 are sometimes called the Kuratowski graphs.

(a)

(b)

Fig. 12.1. Kuratowski graphs. (a) Complete pentagon; (b) Thomsen graph.

REMARK 12.2. Ordinarily, a planar graph is defined as one that can be drawn on the plane with no crossing of branches. However, we intentionally avoided following this convention. The theory of graphs can then be made, at least in the definitions of its concepts, independent of the global topological properties of the Euclidean plane. REMARK 12.3. Although Theorems 12.1 and 12.2 seem to give criteria for the planarity of a graph, it should be noted that discerning whether the conditions of those theorems hold for a fairly complex graph will not be very easy. It will be more practical to examine the validity of the very definition for a graph to be planar; in order to discern whether a given graph is planar or not, we may examine the existence of its dual, i.e., we may construct, e.g., a primitive loop matrix associated with one of its tree-cotree pairs and then examine whether or not there exists a graph that has the same matrix as a primitive cutset matrix. The problem of constructing a graph that has a given matrix as a primitive cutset matrix (or loop matrix) is one of the most important problems in topological network theory. The interested reader should see Tutte [ I ] and Iri [3] (cf. also Appendix 5).

Network-Flow Problems in General

Although we have practically efficient algorithms only for the linear network-flow problems (which will be dealt with in Chapter V), it will be of some use to investigate the properties of network-flow problems (nonlinear in general) and their solutions, without explicit reference to the solution algorithms. Chapters I1 and I11 are the prerequisites for this chapter. 13. CONSTITUENTS OF A NETWORK

We shall use the term “network” to denote a composite concept of a geometrical graph and of physical phenomena superposed thereupon. In the case of a network-flow problem, the physical quantities in question are c c f l o ~and ~ ” “tensions,” and a network is a graph, to each of whose branches a flow r“ (- co < r“ < co)and a tension qK(- co < qK < co)are associated, and each of whose branches is endowed with a function called the “ branch characteristic” which constrains the flow and the tension in a prescribed relation. In the following, we shall sometimes make use of the symbol N ( Q ;C,,. . . , C,) to denote the network whose topological structure is given by graph 8 and whose branch characteristics are CK’s. The roles played by branches deserve attention. The branches of a network contribute to determine the continuity equations for flows and 75

76

1V. Network-Flow Problems in General

tensions in reference to the topological structure of its graph, on the one hand, and, on the other, they possess a physical characteristic constraining the flow and the tension of each branch in a prescribed relation. 13.1. Flow One of the physical quantities (real numbers) associated with a branch b, is called theyow in b, . The flows in a network are assumed to satisfy the continuity condition that the flows do not stagnate, i.e., the sum of the entering (or outgoing) flows at each node is equal to zero; or, in the mathematical expression, if we denote by 5" the flow in branch b, and by DKathe incidence matrix of the graph of the network under consideration, we have (Kh.F.)

f 6,O1~"=0 K=

( a = 1, ..., M ) .

(13.1)

1

We call an n-dimensional vector

6=

n

t"b,

E X(R)

satisfying (13.1) a

K=l

flow configuration. The continuity condition (13.1) is usually called

Kirchhoffs lawforflows (which we shall abbreviate as (Kh.F.)). According to the terminology in Chapter 111, (1 3.1) states that : ayow configuration 5 is a loop. In view of the results in Chapter 111, the same condition may be expressed in various equivalent forms. For example, (13.1) is equivalent to (Kh.F.)

f 0,"t"= 0

( a = 1 ,..., m),

(13.2)

K = l

where D," is a cutset matrix, or to the existence of xp's (p = 1, that

. . . ,k) such

k

(Kh.F.)

t K =C RpKxP

( K = 1,..., n),

(13.3)

p= 1

where RpKis a loop matrix. In Fig. 13.1, the inutitive meanings of the expressions (13.1)-( 13.3) of the continuity of flows are illustrated. 13.2. Tension The other physical quantity (real number) associated with a branch b, is the tension across b,. We associate a quantity, called a potential, with each node. If we denote the tension across branch b, by q, and the potential

77

13. Constituents of a Network

-51

+ t z + 53 = 0

t 4

+ 5 5 - 56

Eq.(13.1)

z

5'

0

Eq.(13.2)

=

,y'

- .y*

Eq.( 13.3)

Fig. 13.1. Continuity of flows.

at node nu by l a ,the continuity condition for the tensions in a network is assumed to have the form M

(Kh.T.)

qK =

1 DKu[,

(K =

1 ,..., n).

(13.4)

a= 1

c q, b, n

We call an n-dimensional vector q

=

K=

E

Y(R) satisfying (13.4) a

1

tension configuration. In the terminology of Chapter 111, (13.4) states that: a tension conjiguration q is a cutset. The continuity condition for tensions is called Kirchhoff's law for tensions (which we shall abbreviate as (Kh.T.)). In view of the results in Chapter 111, (13.4) is equivalent to n

(Kh.T.)

1RpKqK= 0 K=

( p = 1,..., k ) ,

(13.5)

1

where RpKis a loop matrix, or to the existence of ya's (a = 1, . . . , m) such that m

(Kh.T.)

qK =

1 DK"y,

(K =

1,..., n).

(13.6)

a= 1

Intuitively, (13.4) means that the tension across a branch is equal to the difference of the potentials at its two endnodes, whereas (1 3.5) means that the sum of the tensions around a closed path vanishes, as is illustrated in Fig. 13.2.

78

IV. Network-Flow Problems in General

vI = i l

-rz

Eq.( 13.4)

q2

+ 'I3 - v4 - 'Is = 0 Eq.(13.5)

Fig. 13.2. Continuity of tensions and potentials.

The inner product of an arbitrary flow configuration 6 with an arbitrary tension configuration q is obviously equal to zero (Chapter 111, Theorem 9.4) : (13.7) 13.3. Branch Characteristic Corresponding to each branch b, of a network, a characteristic curve C, [which has the properties (i), (ii), and (iii) in $5.2 of Chapter 111 is defined. A pair of flow 5" in and tension q K across branch b, is called an adritissiblcpair, or is said to sutisjy the characteristic C, ,if (tK, q,) E C, . As was shown i n $5.2, two convex functions (to be denoted by (P, and $,) are associated with the characteristic curve C,. The projection of C, to the 5 axis will be denoted by I" and that to the q axis by J , . The interrelations aniong (pK (corresponding to the f in $5.2), $" (corresponding to the g in $5.2), C,, I", and J , were investigated in detail in $5.2, Chapter 11. A pair of flow and tension configurations (6, q) is said to be compatible, or to satisfj all the braticlz characteristics, when (t", q,) is admissible for every branch b, . The (geometrical) reorientation of a branch b, causes the simultaneous change of the signs of 5" and q , , i.e., the rotation of C, by 180" around the origin. We shall give to the branches of special characteristics the special names and the special notations as follows. (a) Current soiirce: The flow in it is constrained to a fixed constant c, whereas the tension across it is arbitrary. The constant value of the flow is called the value of the current source (see Fig. 13.3).

79

13. Constituents of a Network

(a)

( b)

Fig. 13.3. Current source: I" = [c, c], J,

= (-

co, a),qK(c) = 0,@(q)

= CT.

(b) Voltage source: The tension across it is constrained to a fixed constant e, whereas the flow in it is arbitrary. The constant value of the tension is called the value of the voltage source (see Fig. 13.4).

(b)

(a)

Fig. 13.4. Voltage source: I" = (-

00,

co),J,

=

[e, el, q,(()

= e(,

@ ( e ) = 0.

(c) Diode: When it is placed in the same direction as the orientation of the branch as is indicated in Fig. 13.5(b), it has the characteristic as shown in Fig. 13.5(a). When it is placed in the opposite direction as is indicated in Fig. 13.6(b), it has the characteristic shown in Fig. 13.6(a). In any case, a diode imposes the restriction on the sign of flow and on the sign of tension.

(a)

q&)

( b)

Fig. 13.5. Diode (placed in the positive direction): I" = [0, co), J, = 0 (5 2 O), Icr"(T) = 0 (7 2 0).

= (-

co,O],

80

IV. Network-Flow Problems in General

(b)

(a)

Fig. 13.6. Diode (placed in the negative direction): I" = (5 O ) , Wrl)= 0 (rl L 0).

03, 01,

J,

=

[0, m),

V"(0 = 0 (E

(d) Linear resistor: The flow 5" in it is proportional to the tension q K across it. The coefficient of proportion Y in the expression qK = r r is called the resistance of the resistor, and the coefficient g in the expression tK = gqK is called the conductance of the resistor (see Fig. 13.7). Unless otherwise specified, we assume Y 2 0 and g 2 0.

(a)

Fig. 13.7. Linear resistor: I" = (fgrlZ (47 = 1).

( b) 00,

a),J,

= (- 03, m),

~ , ( o&rP,@(?) =

=

In $17 of this chapter and in Chapter V, it will be shown that a composite branch, which is a two-terminal network consisting of branches with the characteristics satisfying properties (i), (ii), and (iii) of $5.2, has again a characteristic satisfying those properties. Postponing the treatment of the general case until there, we shall investigate just below the simplest cases of composite branches. Let us first consider the parallel connection of two branches with characteristic C, and C,, respectively (see Fig. 13.8). The parallel connection, when regarded as one (composite) branch, carries the flow 5 equal to the sum of the flows t1and 5' in the constituent branches [as shown in Fig. 13.8(a)], whereas it has the tension q common with the constituents. Hence, the characteristic curve C of the composite branch can be written as

13. Constituents of a Network

81

(b) Fig. 13.8. Parallel connection of branches.

i.e., for a given value of q we can determine the abscissa of a point on C with ordinate q by drawing the horizontal line with ordinate q, obtaining the abscissae t1 and 5, of its intersections with C, and C, (if they exist) and then summing them. This relation may be abbreviated as 5

c = c ,+ c , .

(13.8)

If we denote the projections of C, C1,and C, on the q axis by J, J , , and J , , respectively, we have J = J , nJ,.

(1 3.9)

It is not difficult to see (and will generally be proved in $17) that, if C, and C, satisfy properties (i), (ii), and (iii) of $5.2, then so does C. The convex function $(q) of the composite branch is related to the functions $'(q) and

82

IV. Network-Flow Problems in General

$'(q) of the constituents by the relation

$(49 = +'(rl) + $,(rl>

(? E J = J ,

J d 7

(13.10)

as is obvious, whereas the relation among the functions cp([), q1(t1),and ( ~ ~ ( 5is, )somewhat more complicated. In order to obtain it, let us fix a 5 belonging to I (the projection of C to the 5 axis). Then there exists a point (5, q ) on C such that (13.11) 5= +

tl t2

with appropriately chosen points ( t l ,17) on C, and Theorem 5.9 (Chapter II), we have Vi(t')

(g2, q) on C, . From

+ $ i ( ~= ) till

(13.12)

for i = 1 and 2. Summing (13.12) for i = 1 and 2, and then substituting (1 3. lo), we have

cpdt')+ c p 2 ( t 2 ) = (el + t%- C$'W + 4%91 = 5rl

-

s maxC5rl - $(41= (P(5),

(13.13)

V E J

where (5.48.2) (Chapter 11) has been taken account of. On the other hand, for any 5' €1' and any 5 , € 1 , such that 5 = 5' 5 , and for any q E J , we have, again by Theorem 5.9,

+

(PI(5')

i.e.,

+ (P2(t2) 2 5v - $(d,

cpl(tl) + cp2(t2) 2 maxctll - $(dl= do.

( 13.14)

$ E J

Therefore, we have

d<)=t; =min 5' + t;'

E

I',

'52. t;2 E 1 2

CV1([')

+ 402(t2)I

C = cPi(t') + V 2 ( t 2 ) ] .

(13.15)

(13.15) can be regarded as defining an operation of forming the new function cp from two given functions 'pl and c p 2 , and cp is sometimes called the miniinurn conoolute of 'pl and 'p, and is denoted by cp = 4% O V 2 .

(13.16)

Evidently, 0 is commutative and associative. Summarizing, we have Theorem 13.2. The maximum transform of the minimum convolute of two convex functions is equal to the sum of their maximum transforms.

13. Constituents of a Network

83

REMARK 13.1. It is interesting to compare the foregoing theorem with the famous theorem in the theory of Fourier transforms that the Fourier transform of the convolute of two functions is equal to the product of their Fourier transforms. Similarly or, more exactly, dually, let us consider the composite characteristic of the series connection of two branches (Fig. 13.9). As is shown in Fig. 13.9, the flow 5 in the composite branch is equal to each of the flows and 5' in the constituent branches, whereas the tension across the composite branch is equal to the sum of those ql and q2 across the constituent branches. The composite branch characteristic C is therefore determined from the characteristics C1 and C2 of the constituent branches as

r'

c = ((5, 'I)I 'I = + 'I1

y/2

9

(5, 'Ill E c,,(r, 'Id E c,>.

(b)

Fig. 13.9. Series connection of branches.

(13.17)

84

IV. Network-Flow Problems in General

The relation is abbreviated as 9

c = c ,+ c , .

(13.18)

Other properties we described for the parallel connections hold mutatis mutandis also for the series connections. The foregoing arguments can easily be generalized to the case of the series or parallel connection of more than two branches. (The reader should explicity write them out.) Now we are at the position to consider the characteristics of the branches formed as the series-parallel connections of (a) current sources, (b) voltage sources, and (c) diodes. (e) Current limiter: The parallel connection of a current source of value c and a diode such as shown in Fig. 13.10(a) has the characteristic of Fig. 13.10(b). A composite branch of this kind will be called a current limiter (with value c ) and denoted in abbreviation as shown in Fig. 13.10(c).

e

-ID--

5"

(a)

Fig. 13.10. Current limiter: I" = (-

q (77 2 0).

(b) CO,

c ] , J,

(c) =

[0, a),yK(,(5) =0

(e 5

c), p(7)=

(f) Voltage limiter: The series connection of a voltage source of value e and a diode such as shown in Fig. 13.11(a) has the characteristic of Fig. 13.11(b) and will be called a voltage limiter (with value e).

13. Constituents of a Network

85

(g) Single-step step-function : We often encounter the branches of the types shown in Figs. 13.12 and 13.13.

(h) General step-function: Any branch with a monotone nondecreasing step-function characteristic can be composed as a series-parallel connection of branches with single-step step-function characteristics. For

86

IV. Network-Flow Problems in General

example, the branch with the characteristic of Fig. 13.14(a) can be composed as the parallel connection of the branches of the kind Fig. 13.12 [see Fig. 13.14(b)] or as the series connection of the branches of the kind Fig. 13.13 [see Fig. 13.14(c)]. It can be composed in many other ways, of which two are illustrated in Fig. 13.14(d) and (e).

-t---

(e)

(d)

Fig. 13.14. Step-function.

The foregoing example will suggest the following theorem, of which the formal proof is left to the reader.

Theorem 13.2. Any branch with a monotone nondecreasing stepfunction characteristic can be composed as a series-parallel connection of tic current sources, n, voltage sources, and n, diodes, where n, is equal to the number of vertical segments of the step-function, n, to that of horizontal segments, and n, to that of corners. The numbers n,, n,, and t i D satisfy the relations

In, - n,l 5 1,

n,

= n,

+ n, - 1.

(13.19)

In concluding this section, we note that any branch with a monotone nondecreasing piecewise linear characteristic curve can be composed as a

15. Minimization Problems on a Network

87

series-parallel connection of current sources, voltage sources, diodes, and linear resistors. However, we shall omit the details about such branch characteristics. (The reader should work out the constructions analogous to those in Fig. 13.14.) 14. REMARKS ON DUALITY

The reader will probably have noticed that all the descriptions (except for those including the concept of nodes) in regard to networks in $13 have a perfectly dual structure, which is a natural extension of what we noted in Chapter 111 (see $10) for the topological properties of graphs. The dual correspondence of the graphical or topological concepts that is tabulated in $10 may be augmented with the following tabulation to involve also the network concepts already treated and to be treated in this chapter: flow current source current limiter (Kh.F.) diode resistor

tension voltage source c-) voltage limiter ++ (Kh.T.> c-) diode c-) resistor etc. ++

t ,

Almost all the descriptions in this chapter will be put in the dual structure. Although we shall not point out everywhere which statement corresponds to which in the framework of duality, the reader should always bear it in mind. Moreover, we shall sometimes omit the proof to a theorem whose dual has already been proved. In such a case, the reader is asked to supplement the omission or, at least, to convince himself of the possibility of doing it. 15. MINIMIZATION PROBLEMS ON A NETWORK

The problem of obtaining a compatible pair of flow and tension configurations on a given network can be expressed in the three seemingly different forms to follow (Problems 1-111). This section is devoted to the formulation of these problems as well as the discussions on their interrelations. By a “ network-flow problem,” we shall mean a problem of the kind to be treated in this section. We assume also that the specifications to the given network N ( S ;C,,. . . , C,) are the same as those given in $13.

88

IV. Network-Flow Problems in General

15.1. Problem I: Minimization Problem in Terms of Flows

c t"b, n

To obtain a flow configuration

6=

[E X(R)]

which, satisfying

,=l

the condition

5" E I"

for each branch b,

(IC = 1,

. ..,n),

(15.1)

minimizes

@D(t>

n

K=

(15.2)

qK(<">,

1

where, as in $13, I" is the projection of the characteristic curve C, of the branch b, to the 5 axis, and q,(5) is the associated convex function. The function @(g) is called the objective function of Problem I. This problem can be regarded as a rigorous formulation of the transportation problem roughly delineated in $1 of Chapter I. A flow configuration that satisfies (15.1) (and (Kh.F.), of course) is called a feasible $ow configuration, and a solution of the problem an optimal $ow conjiguration. The feasible flow configurations form a closed convex subset I of V(R, B), since we have I = (I' x x I") n X(R) (15.3)

---

with closed intervals ZK's and a linear subspace X(R). The set of optimal flow configurations will be denoted by S , . 15.2. Problem 11: Minimization Problem in Terms of Tensions

c n

To obtain a tension configuration q =

Y=

t,~, b, [E Y(R)]

which, satis-

1

fying the condition q,

E J,

for each branch b,

(K =

1,

. . . ,n),

(15.4)

minimizes

where, as in $13, J , is the projection of the characteristic curve C, of the branch b, to the q axis, and II/,(q) is the associated convex function. The function "(q) is called the objective function of Problem 11. This problem can be regarded as a rigorous mathematical formulation of the project scheduling problem roughly delineated in $2 of Chapter 11. A

89

15. Minimization Problems on a Network

tension configuration that satisfies (15.4) (and (Kh.T.), of course) is called a feasible tension configuration, and a solution of the problem an optimal tension configuration. The feasible tension configurations form a closed convex subset J of V(R, b): J

= (.I1 x

.

x

.In)n Y(R).

(15.6)

The set of optimal tension configurations will be denoted by S,, . It is important to note that Problem I and Problem 11, as we have defined, are mutually dual of each other.

15.3. Problem 111: Equilibrium Problem in Terms of Flows and Tensions Our third problem contains flows and tensions in a symmetrical form.

To obtain a compatible pair of flow configuration 6 =

n

2t"b, I(=

n

q, b,

and tension configuration q =

[E

[E

X(R)]

1

Y(R)], i.e., a pair such that

?C=l

(r,q,) E C,

for each branch b,

(K = 1,

.. .

n),

where, as in §13¶ C, is the characteristic curve of the branch b,

(15.7)

.

This problem is none other than the problem of a nonlinear resistor network mentioned in $3 of Chapter I, flow and tension being compared to electric current and voltage, respectively, and C, to the current-voltage relation of a branch. The set of solutions (6, q) of the problem will be denoted by S,,,. 15.4. Relations among Problems 1-111

What we shall deal with in the following may be regarded as a special case of the Kuhn-Tucker theorem on nonlinear programming. However, we shall develop our theory without reference to the Kuhn-Tucker theorem. Due to the special characters of the problems we are dealing with, we can go much farther than the general theory could. It is also hoped that a full understanding of a special case will serve to help the better understanding of the general case. Let 6 (E I) be an arbitrary feasible flow configuration and q (E J) be an arbitrary tension configuration. The function Q(5,q)

= W) + Wrl) =

i

K= 1

CV,(t")

+ ICl"(t1,)l

(1 5.8)

of 2n variables is continuous on the closed convex set I x J, and, further-

90

IV. Network-Flow Problems in General

more, since the q K 7 sand GK's are convex, R, as well as and Y , is also a convex function (Theorem 5.2). It follows from Theorem 5.9 ($5.2 of Chapter 11) and (13.7) that

'(6,

n

11) 2

rKr?K K=

If

(6

=

" 9

(c, 4) is a solution of Problem 111, i.e., if (c,fi) qK(e")

for all

K,

q

(15.9)

J>'

1

+

GK(8K)

=

E

S,,,, then (15.10)

Z K q K

so that we have Q(f,

fi) = 0.

(15.11)

Conversely, if (15.1 1) holds for a $ E I and an fi E J, then (15.10) must hold for all K , so that, by virtue of Theorem 5.9, (15.7) must hold for all K. Hence, fi) E S,,,. Thus, we have

(c,

Theorem 15.1. If neither I nor J is empty, i.e., if both Problem I and Problem I I have a feasible configuration, then Q(6, q) is nonnegative on I x J. Furthermore, we have

s,,,= { k q ) I Q ( L q) = 0, (G? s)E I x J > .

( 15.1 2)

(c,

Let fi) E S,,, be a solution of Problem 111. Then, for an arbitrary feasible flow configuration 6 (E I), we have

@(5)= '(6,fi)

- Wfi)

L Q(6,fi)

- Wfi) =

w, (15.13)

and, dually, for an arbitrary feasible tension configuration q (E J),

Wq) = '(S, q) -

we) 2 Q(t,fi)- we) = Vfi).

(15.14)

Hence follows

(6,

Theorem 25.2. If fi) ES,,,is a solution of Problem 111, then f is an optimal flow configuration, i.e., a solution of Problem I , and fi is an optimal tension configuration, i.e., a solution of Problem 11. In other words,

s,,,c= s, x s,,.

(15.15)

The following theorem is the most fundamental in this context. Theorem 25.3. If Problem I has a solution 6, then there exists a solution fi of Problem I I such that 6) is a solution of Problem 111. Dually, if Problem I 1 has a solution Q, then there exists a solution $ of Problem 1 such that fi) is a solution of Problem 111.

(e,

(c,

91

15. Minimization Problems on a Network

PROOF. We shall give the proof to the first half of the theorem because the rest can be proved dually. Let us denote by 23, the set of branches for which g" is an interior point of I", by 23, the set of branches for which is the left end of I", and by 23, the set of branches for which g" is the right end of ZK(23 = 23, u 23, u 23,). For another feasible flow configuration 5 (E I, not necessarily optimal), we introduce the expressions

s"

E

A t K= 5" - g.

(E

> 0, b,

( 15.16)

E 23)

and A t K= A'tK - A-5"

(b, E

B,),

At''

(b,

SRn BLA

=0

where A'CK 2 0 and A-5" in terms of A*tK's as

c D," Atu

=

2

E

2 0 for all K. The feasibility of 6 is expressed D,"A-5" = O

D,"AfrU -

bKE!&U(m~-5~)

(a

= 1,

..., m),

b,EBru(8~-5~)

A'tK 2 0

[b,

E

231 u (23, - S,)],

A-5" 2 0

[b,

E

2 3 1 u (23, - BL)],

(15.18)

where D," is a (primitive) cutset matrix of the graph of the network. Making use of the notations cpL(*) introduced in $5.2 for the one-sided which means derivatives and also of the notation [XI', [XI'

=x

if x 2 0 ,

=O

if x 5 0,

(1 5.19)

5 as @(5>- @(5>= @tS + E AS) - @(S) we can express the optimality of

(15.20)

92

IV. Network-Flow Problems in General

where O ( E ) denotes a quantity such that account of the relation

O(E)/E

tends to zero as E does. On

[A*<" - ATtK]' = A'5" - min(A+<", A-t"),

(15.21)

(15.20) is rewritten further as

c

@(k) - @(t) = E( bKEBIu('&-'%d -E

A'C" -

cp;(+)([")

c

q;(-)([")A-t")

b,€%U(%x-&.)

C {q:!')(p)- cp:!-)(p)} min(A+5", A-5") + O ( E ) b, E

2 0.

(1 5.22)

Therefore, recalling that

-

cp"!(+) (5")

2 cp;(-)(?>,

we have

cp;(-)(p") 8-5")+ O ( E ) 2 0.

A'5" -

cp:(+)(5^") '(b,EBI

323-Z)~)

b,

E BI U

(BR-SL)

(15.23) Hence, for an arbitrary set of A'5"'s and A-<"'s satisfying (15.18), if we choose E sufficiently small, we must have

C

cpL(')(p)A'<"

b i E B I u (BL-BR)

b,

E

C cp;(-)(p)A-Y 2 0.

(15.24)

BI u ( B R - ~ L )

t

Thus we have seen that, for to be optimal, it is necessary that (15.24) holds for an arbitrary set of A'5"'s and A-<"'S satisfying (15.18). (15.18) can slightly be modified to give the following equivalent conditions :

C

DK(IA'5"-

b, E BI u (BL-BBH)

C

D,"A-5"20

( a = l , ..., m),

b,E@ru(Bn-B~)

(15.18.1)

-C ~,EBIu(BL-BR)

+ C

D," A'<" b,

E '51u

D," A-t" 2 0

(a = 1,.

..,m),

(BR-BL)

(1 5.18.2) A'C" 2 0

[bK E 8 1 u (8,- BR)],

(1 5.18.3)

A-5" 2 0

[b, E 8, u (23, - BL)].

(15.18.4)

In view of Farkas' theorem (see Appendix I), we can assert, from the fact that the inequalities (15.18.1)-(15.18.4) entail the inequality (15.24), the existence of nonnegative variables y,', y a - , wK', w,- [corresponding

93

15. Minimization Problems on a Network

to (1 5.18.1)-( 15.18.4), respectively] which satisfy the relations

(15.26) (15.25) means that

I(+) V K

(5 ) > = 9, 9, 2 4:(-'(P") 9, : arbitrary n

(15.26) states thatq

=

3, b,

(bK

L '

(bKE8,

- %R)> - 'L),

(15.27)

(b, E BLn BR).

is a cutset, and (15.27) states that (g", Q,)

E

K = l

C, for all K. Therefore, the given 5 and the .fi thus determined constitute a solution (5, 4) of Problem HI. Q.E.D. Theorem 15.4. If 5 is a solution of Problem I and fi a solution of Problem 11, then (f, 9) is a solution of Problem 111. In other words,

s,,,2 s, x % I *

(15.28)

PROOF. Since 5 is a solution of Problem I, there is a solution 4 of 4) is a solution of Problem I11 (Theorem 15.3). Problem I1 such that From Theorem 15.1, it follows that

(5,

ncf,4)= Q(5) + Y(4) = 0.

(15.29)

Since fi is also a solution of Problem 11, we have

W.4) = W4).

(15.30)

From (15.29) and (15.30), we have

Q(5,CI) = Q@ + W4) = 0, so that

(5, -4) is a solution of Problem I11 (Theorem 15.1).

(15.31) Q.E.D.

94

1V. Network-Flow Problems in General

I t has been seen that Problems I , I!, and 111 are equivalent to one another in the sense that, if one has a solution, then so do the others (Theorems 15.2 and 15.3). Furthermore, we have from Theorems 15.1 and 15.4 (1 5.32) SII, = SI x SI,.

So much for the case where the problems have a solution. Let us then investigate the other possible cases by classifying them according to the existence or nonexistence of feasible configurations. All the possible cases are classified into the following four mutually exclusive cases : (a) 1#0, J Z 0 ; ( 4 I=@, JZG;

(b)

I#@,

J=@;

(d) I = 0 , J = 0 ,

where I or J is, as before, the set of feasible flow or tension configurations. These four cases actually take place as is illustrated in Fig. 15.1. [The reader should verify that the four networks in Fig. 15.1 correspond to the preceding four cases (a)-(d).] Before entering the detailed investigations into the preceding four cases, let us establish a lemma for later use.

<>$ 6 % )

e>O

(C)

(d)

Fig. 15.1. Four kinds of networks.

Lemma 15.1. If a)@ has a lower bound on I (nonempty), then it attains the minimum on I, i.e., Problem I has a solution. If Y(q) has a lower bound on J (nonempty), then it attains the minimum on J, i.e., Problem I1 has a solution.

PROOF. The infimum, i.e., the greatest lower bound of @(Q on I, whose existence is assumed, will be denoted by ( 1 5.33)

95

15. Minimization Problems on a Network

Due to the definition of the infimum, there is a sequence of feasible flow configurations 6,6, . . . (E I) such that 0 1

lim

~ ( 6 =) 6.

i-m

(15.34)

i

If I is bounded, i.e., if it is compact, then we can choose a subsequence of 6's having the limit f for which i

~ E I .

and

O(f)=6

(15.35)

The case where I is unbounded requires further consideration. Let the set of branches 8 be partitioned into three subsets B,,B,, and B3, where 23, is the set of those branches bK's for which the sequence {r,5", . . .} has a point of accumulation, 23, is the set of those branches 0

1

b,'s for which the sequence {Y, 0

Y, . . .} has no upper bound (or no lower 1

bound) but for which J , is bounded upwards (or downwards), and 23, is 8 - (23, u 8,). Taking an appropriate subsequence if necessary, we may assume that the sequence (6, 6,.. .} has the following properties, where E 0

1

and 6 are arbitrary (fixed) positive numbers and n, is the number of branches of 8,: (1) For every branch b, E B,, Iim 5" = i-co

5" 5 5" 5 5" 5 ... 0

1

and

2

2-'c/n1 > 15" - PI,

for any

and

4" between P and r". 0

E

B2,

15" - <"I > E , i

lim 5" i+m i

and

= 00

(15.363

i

Icp,(g") - cp,(g")l < 6/nl (2) For every branch b,

eK,

i

(15.37)

i-1

(resp. lim i-w

< " = - a), i

96

IV. Network-Flow Problems in General

(3) For every branch b,

E

B3,

0 < r" < 5" < * * * 0

(resp. 0 > 5" > 5" > 0

1

me-),

1

15" - 5"l > E, i i-1 5" = co

lim

(resp. Iim

i+m i

(15.39)

5 ,= - co),

i+m i

and

(15.40) [Here it should be noted that property (iii) in 55.2 (Chapter II), which we assume is possessed by every branch characteristic, plays an important role in the foregoing partitioning of S.]Next, let us consider the equisignum decomposition of 6 - 6 into minimal loops (see $9.1 and Appendix 2): i

6-5= i

O

0

(xij > 0; ryj = 0, 1, or - l),

xij rij

(15.41)

j

where the rii)s are elementary loops. Although the decomposition is not unique, there can appear, in any decomposition, no elementary loop rij such that (15.42) car rij s 23, , C a, r; < 0; K

for, if such an rijo appeared, the flow configuration

6 + xrijo would

be

i

feasible (i.e., have

E I)

for any positive x, and, by virtue of (15.38), we should

1

+ x r i j o ) = @(t> +x C i

r:jo

9

K

which would, however, contradict the assumption that @(6) is bounded downwards on I. Therefore, in a decomposition such as (15.41), if the carrier of an elementary loop rij is included in S 2 , the sum of the a,'s around it must be nonnegative : a, rrj 2 0 K

It is also obvious that, for every i,

C' x i j < E , i

(1 5.43)

15. Minimization Problems on a Network

where

c'

97

means the summation over all those j ' s

j

car rij n bl#

0. We split the summation

c' for those ri;s

in (15.41) into three, i.e., j

for which car rij n b, #

i

which car rij s b2,and and car r i j n Sj3# we have

those ri;s

c"'for those rii)s for which car rij i

0.From (15.36)-(15.40)

i

0,c" for i

2 cp,(t")

(PK(5")

for which

n 23, = 0

and the convexity of for

b,

E

b,,

uK

for

b,

E

23, ,

C,

for b,

E

b,,

-

for

(P,'S,

0

cp,(rK) = cp,(r") i

0

ij 1

+ C xijr;

~ ~ i( 52" c~,(r") ) 0 +

(15.44)

where

C

rrjc, >

for every b,

lall

E

b, such that

i-rj

# 0.

(15.45)

b r E 232

Therefore, we have

(1 5.46)

or, setting u =

c

bx E ' 8 2

0

= min

bx E 'Bi

1 4 1+

(

ICKI

c Ic,l c

(> 01,

bK E 23s

(15.47)

lull]

-

bn E 232

(>Oh

and taking account of (15.43) and of the obvious relation

we have

@(t)2 @(g) - 6 - EU i

0

- EV

+ min b,E'Bo

15" - Ylu. i

0

(1 5.49)

98

IV. Network-Flow Problems in General

The relation (15.49) is now independent of the manner of decomposition (l5.41),and, in addition, 6, as well as u and o, is independent of i. (15.49) indicates that the sequence {@(k), @(E,), . . .} would diverge to the positive 1

0

infinity if B, were nonempty [cf. (15.39)], contradicting (15.34). Thus we can conclude that B3 is empty. We then consider to modify each 6 into = i

5 + AG in i

i

such a way that

i -

^

tK = i" i or

AtK = tK- <"

for every

b,

E

8,.

(15.50)

i

i

In order to do this, we choose a tree-cotree pair (2, 'L) such that T contains as many branches of 8,as possible, denoting the associated primitive cutset matrix by D,".The condition that be a loop is

5 i

C

D," AtK = 0

bieS

(U

= 1,.

.., m).

(15.51)

i

If 8,n 2 # 0, i.e., if there is a branch of Bl belonging to the tree 2, then the cutset corresponding to that branch-let it be, say, the aoth of the primitive cutsets-consists of the branches of dl only (for, otherwise, we could choose another cotree consisting of more branches of Bl than z). For such a cutset (15.51) is of the form (15.52)

Since D",", b,E'RI

=O

foreach i ,

(15.53)

i

it follows from the first relation of (15.36) that ( 15.54)

Hence, subtracting ( 1 5.53) from (1 5.54), we see that (1 5.52) is identically satisfied by the A y ' s specified by (15.50). Let us further require that I

AtK = 0

for every b, E 8, n T.

(1 5.55)

i

Then the rest of the A<"'s, i.e., those corresponding to B2 n 2, are

15. Minimization Problems on a Network

99

uniquely determined by (15.51) (cf. also Theorem 9.9 in Chapter 11, $9.4),and we have, from (15.36), (15.51), and (15.55),

IAt"l < 2 - i ~

for every b, E B2 n 2.

(1 5.56)

i

(15.55) and (15.56)combined with (15.36)and (15.37)show that

? E IK

for every i ( = 1,2,. . .) and every K.

(15.57)

i

Therefore, f ' s are feasible, i.e., i

for all i (= I, 2,. ..).

5E I i

(1 5.58)

Since

c

!im

~ + m b,~(81

%A<")

c

=

cp,(r"")

=

i

blCE(81

I

brcE(81

4oK,

and

(15.59) i

E (82

I

- ( ~ ~ ( 5 "=) )

{40,(?) b,

i

I

1

aKA5K 5 2 - i ~ i

/blC.s%

b,~9)2

bKlr

we have lim

a([>= lim ~ ( 5 =) 4.

i+co

i

(15.60)

i

i+co

The equisignum decomposition of f - f into minimal loops i

1

5 - 5 = c Zij Pij i

l

(15.61)

j

will yield the relations, similar to (15.44)and (15.46),

cp,t5"")

= cp,(?)

=

i

cp,(?)

= i

cp,(P)

for

b, E B , ,

1

( ~ ~1( 5 "+" )

(1 5.62)

(7

for b,

giji ;t)u,

E

23,

,

and

@(5) i

= a([) 1

+ c" j

1

Iij i;&u,, brE82

(15.63)

where it is taken into account that every Pij in (15.61)has its carrier in B 2 .

100

IV. Network-Flow Problems in General

It is shown as before that, for every iij,

(15.64) Therefore, we have

(1 5.65) From (15.60) and (15.65), it follows that

However,

lim

@(El)

i-tm

i

=

6 2 @(El.

(15.66)

1

6 is the infimum of @(t)on I, so that we finally have

@(E) = 6,

(1 5.67)

1

6 being an optimal flow configuration, i.e., a solution of Problem 1. 1

The second half of the theorem can be proved similarly.

Q.E.D.

By means of the foregoing lemma, we can now see the details of cases (a)-(d). In case (a), since there are 5 E I and q E J, we have, for an arbitrary 0

0

5 €1, or

W) 2

(15.68)

-Wq). 0

This means that

@(t)has

a lower bound -Y(q) on I, so that Problem I n

has a solution. Hence, by Theorem 15.3, Problem I1 as well as Problem I11 has also a solution. In case (b), @(g) can have an arbitrarily small value on I, for, otherwise, Problem I would have a solution (Lemma 15.l), and, consequently, Problem I1 would have a solution (Theorem 15.3), in contradiction to the assumption J = 0. Case (c) is the dual of (b). The foregoing considerations may be summarized in the following theorem. Theorem 15.5. For a given network-flow problem, one and only one of the following four cases takes place:

101

16. General Properties of Solutions

(a) There exist both a feasible flow configuration and a feasible tension configuration. In this case, all three problems, i.e., Problems I, 11, and 111, have a solution. (b) There exists a feasible flow configuration but not a feasible tension configuration. In this case, the objective function @(G) of Problem I is unbounded downwards on I. (c) There exists a feasible tension configuration but not a feasible flow configuration. I n this case, the objective function "'(1)of Problem I1 is unbounded downwards on J. (d) There exists neither a feasible flow configuration nor a feasible tension configuration. 16. GENERAL PROPERTIES OF SOLUTIONS

The definition of a network-flow problem was given in the preceding section, and the practical algorithm to obtain the solutions will be dealt with in the following chapter. In this section, the general properties of the solutions as well as the conditions for the existence of a solution are investigated. 16.1. Convexity The solutions of Problem I form a closed convex set S, in V(R, d), and those of Problem I1 a closed convex set S,, in V(R, d),as is evident from Theorem 6.1 (Chapter 11, $6). The set of solutions of Problem 111, i.e., S,,I,is also closed and convex in V(R, 23) x V(R, 23) due to (15.32). 16.2. Uniqueness The following Theorem 16.1, although it is rather trivial itself, is useful and powerful for the investigation of the approximation and uniqueness problems.

e,

Theorem 16.1. Let 9 be a solution of Problem 111 for a network N ( 8 ;C,, . . . , Cn).Then, it is also a solution of Problem 111 for any network N ( 8 ; C,', . . . ,C,') which has the same topological structure 8 as the original network but which has different branch characteristics Ci', .. . , C,' such that

(p,9,) E c,'

for every K.

(16.1)

From Theorem 16.1 follows the important uniqueness theorem.

102

IV. Network-Flow Problems in General

Theorem 16.2 (uniqueness theorem). Let

(5,q) and (g, q) be two solu1

2

1

2

tions of Problem I l l for a network N ( 6 ;C,, . . . ,C,,).Then, for each branch b, , either 4" = 5" or qk = 7". Consequently, if there is a cotree on 6, the 1

2

1

2

characteristic curves of whose branches have no horizontal segment, then the solution of Problem I is unique; similarly, if there is a tree on 8, the characteristic curves of whose branches have no vertical segment, then the solution of Problem 11 is unique. PROOF. Since S,,, is convex (25 + (1 - A)G, l q 2

1

+ ( 1 - l ) q ) is also a

1

2

solution of Problem 111 for an arbitrary l between 0 and I . Therefore, for an arbitrary 2 between 0 and I ,

(At" + (1 I

- AX", 2

h, + (1

- 4q,) E

1

2

c,

for ii = I , . . . , / I .This means that each curve C, must contain a straight line segment connecting (th,q K )and q K ) .However, by virtue of Theorem I

(c", 2

1

2

16.I , both (5,q) and (5,q) remain to be solutions of Problem I11 if we replace 1

2 2

1

C, by C,' such that both (tK, qK) and 1

and 1 1 > ~ q K for some 7 -

ii,

1

(c", q,) 2

are on C,'. If we had

2

5" > 5" 2

1

then we could choose, in many ways, a C,' such

1.

that it satisfies properties (i)-(iii) in $5.2 but that it is not straight between (t",qy) and (t",qK). Hence, we should have 5'' = tKor q, = qK (cf. Fig. 1

1

2

2

1

2

1

2

16.1). The remaining part of the theorem follows from Theorem 9.9 (Chapter 111, $9.4). Q.E.D.

Fig. 16.1. Proof of the uniqueness of solution.

103

16. General Properties of Solutions

16.3. Nonamplification and Reciprocity

We shall investigate the counterparts, in network-flow problems, of the nonamplification and reciprocity properties in linear resistor networks treated in Appendix 3. As we shall see, the nonamplification is related to the monotonicity of the characteristic curves of branches and the reciprocity to their smoothness. Let us begin with the nonamplification properties. We suppose that branch b, is a current source of value cK and that the value of the flow in = 1 or IC # 1) is ta.Our problem is to investigate the variation branch b, (IC in 5, as cK varies, with all the other branch characteristics fixed. We have also a dual problem. Since we cannot expect in general the uniqueness of the solution in the strict sense (cf. Theorem 16.2), we have to content ourselves with Theorem 16.3 (nonamplijkation theorem). Suppose that Problem I for a network Nl E N ( 6 ; C l , . . . , C,,, . .., C,) has a solution 6 where branch b,, is a current source of value cXo. Then, if Problem I for the . . . , C,) has a solution (where branch network N 2 = N ( 6 ; C1,. .., b,, is a current source of value cK0f AcKo, all the other branches having the same characteristics as the corresponding ones in the original network), it has a solution 5 A6 such that

e,,,

+

1At"I I IAcK0I

for every K.

(1 6.2)

Dually, jf Problem I1 for a network N ( 6 ; C,,..., C,,, ..., C,) has a solution where branch b,, is a voltage source of value e,, and if Problem I1 for the network N ( 6 ;C1,. . ., C,,, . . ., C,) has a solution (where branch b, is a voltage source of value e,, &a,, all the other branches having the same characteristics as the corresponding ones in the original network), then Problem I1 for the latter network has a solution q Aq such that

+

+

for every 1.

1Aqa1 I IAe,,l

(16.3)

PROOF. We give a proof to the first half of the theorem. The latter half will be proved in the dual manner. Let 5 = 6 be a solution of Problem I for 1

the original network and 6 be one for the modified network, and let q and q 2

1

2

be the corresponding solutions of Problem I1 (cf. Theorem 15.3). For each IC ( # K,,), we denote by C,' the straight line passing the two points (t",q,) and

(t",q,) 2

2

1

1

if they are different points, or the vertical straight line passing

104

IV. Network-Flow Problems in General

(t”,q,) if they coincide with each other. Then we consider the networks 1

1

Nl’= N ( 6 ;Cl’, . . . , C:o-l, C,,, C:,,,, . . . , C,’) and N 2 ’= N ( 6 ;C1‘, ..., C:o-l, C,,, C:o+l, . . . , C,’), where C,, and C,, are the characteristic curves of the current sources of values cK0and cK0+ AcK0,respectively. As is obvious from Theorem 16.1, 6 and 6 are solutions for Nl’ and N2’, 2

1

respectively, with tKo = cK0and 5““ = cK0+ AcK0.In considering the equa2

1

tions to be satisfied by A6 = 6 - 6 and Aq = q - q, we may assume that 2

1

2

1

none of C,”s (K # K ~ are ) vertical, since we may open-circuit in advance those branches with vertical straight-line characteristic curves if such exist. A6 is a loop (since both 6 and 6 are loops), Aq is a cutset (since bothq andq 1

1

2

2

are cutsets), and, for every K ( # K,,),

4, = r,A5r, where rKis the slope of C,’. Thus, the equations to be satisfied by the A r ’ s and AV,’S are those of a linear resistor network, say N 3(cf. Appendix 3), where b,, is regarded as the only current source of value AcK0,and the other branches are regarded as linear resistors (if r, > 0) or as voltage sources of value 0 (if r, = 0). If the set of branches with r, = 0 is loop-free, then, by Appendix 3, we can conclude that

1Ar1 I IAcK0I

for all

K.

If it is not loop-free, we proceed as follows. We classify the elementary closed paths consisting of branches with null resistances into two classes (cf. Fig. 16.2): (a) The incremental flows AtKaround one of such closed paths are in the same direction (i.e., either the A r ’ s in the branches lying in the positive direction in the closed path are positive, whereas those in the branches lying in the negative direction are negative, or the AtK’sin the branches lying

(4

(b)

Fig. 16.2. Elimination of a null-resistance closed path.

105

16. General Properties of Solutions

in the positive direction are negative, whereas those in the branches lying in the negative direction are positive). Since r, = 0 means q, = q , , the part 2

between

1

(r,q,) and (r,q,) is a horizontal straight-line segment of C, 1

2

1

2

as well as of C,'. Let us denote the elementary loop corresponding to the elementary closed path by p, and set (16.4)

where (16.5)

and the positive or negative sign is adopted in (16.4) according as the nonnull p K AtK's(which are of the same sign by assumption) are negative or in the branches not conpositive. Then the t'"s are the same as the

r's

2

2

tained in the closed path, whereas, in each of the branches in the closed path, t'" lies between t" and 5". Hence, (t'",q,) is on C, as well as on C,' 2

for every IC ( # K

2

1

~ and )

(IKo

2

=

t

+

2

2

= cK0 AcK0,so that

2

(S', q) is also a solu2

2

tion of Problem 111 for Jtr2 as well as for N2'. Moreover, in at least one of the branches in the closed path, we have

Therefore, if we take

6' for 6 , the corresponding linear resistor network 2

2

N 3 will have fewer elementary closed paths of the kind now in question. In this way, we can obtain a solution 6 of Problem I for N2such that the 2

corresponding linear resistor network N 3 has no elementary closed path that consists of branches with null resistances and around which the incremental flows A F are directed in the same direction. Thus, we have only to consider the following case (b).

Ar

around each of the elementary closed (b) The incremental flows paths consisting of branches with null resistances are not all in the same direction. Let us take up one of such closed paths and suppose that branch bAocarries the incremental flow AtAoof the maximum absolute value around the closed path. Denoting by p the corresponding elementary loop, we set (16.6) AS' = AS f (A~")P, where the plus or minus sign is taken in (16.6) according as pAO= - 1 or

106

IV. Network-Flow Problems in General

pAO= 1. Obviously, A t ‘ is a loop, and (At‘, Aq) is also a solution for the

linear resistor network N 3 .[Note that

(6 f (AtAo)p, q) is a solution for 2

2

N 2 but ’ is not necessarily a solution for A’, .] Since At’“ = A t “ in a branch b, not belonging to the closed path now in question and since there is, by assumption, a branch, say b, belonging to the closed path such that

1At’”l = 1AC”l

+ IA<”l

5 lAtAol

(16.7)

and, consequently, that, for every branch b, belonging to the closed path,

IWI 2 lAtL0l 2 lAtAl,

(16.8)

it suffices to prove that IAt’,l

5 IAcrol

for all K.

(16.9)

Since A(”O = 0 by (16.6), (At‘, Aq) is a solution for the linear resistor network .N3’which is obtained from N 3 by open-circuiting branch ,b, . Thus, the nonamplification property (16.2) of the given network is reduced Network N 3 ‘obviously to that of N 3 and then to that (16.9) of N3’. has fewer closed paths consisting of branches with null resistances than N 3 has. Repeating in this way, we can get a linear resistor network with no closed path of null-resistance branches, in which the nonamplification Q.E.D. property is proved to hold. The other property of fundamental importance is the reciprocity, which may be stated as follows. Theorem 16.4 (reciprocity theorem). Let {, fj be a solution of Problem 111 for a network N ( 0 ; C,,.. ., C,) such that for every K, C, has a well-defined tangent at (tK,fi,) which is neither horizontal nor vertical. Furthermore, for a specified pair of branches b, and b,, let us denote by :

+ AqAo the tension across branch b,, when a current source of value AcK0is attached in parallel with b, , fi,, + Aq,, the tension across b,, when a current source of value AcAo is attached in parallel with b,, [ l o + AtAo the flow in b ,, when a voltage source of value Ae,, is inserted in series with b, and tKo+ A t K o the flow in b,, when a voltage source of value Ae,, is 9,,

inserted in series with ,b,

.

107

16. General Properties of Solutions

Then, at the limit of AcK0-,0, AcAO-+0, Ae,, -A?Ao

A?,, --

-,0,

and AeAo--* 0, we have

AtKo AtAo

-- --

and

AcK0 AcAO

AeA,

(16.10)

Ae,,.

PROOF. By assumption, the convex functions (P, and I,?, associated 9,). Attaching in parallel a current with C, are twice differentiable at into $""(?) + (AcK0)q. The conditions source of value AcK0changes that the p's and 9,'s satisfy are the continuity conditions and

(p, vo(q)

I,F'(Q,) = 4"

for every K.

(16.11)

The solution of Problem I11 for the network with the current source attached in parallel with b,, is uniquely determined by virtue of Theorem 16.2, which A&, 4 Aq). The (g' AF)'s and (9, + A?,)'s we shall denote by satisfy, besides the continuity conditions,

(e +

$,'(QK

$""'(9,,

'?K)

+

=

+

4" + A t K

for

K

( Z KO),

(16.12)

+ A?,,) + AcK0= Po + A?.

From (16.11) and (16.12), we have $'"'(&JAq, F(q,,)A?,,

= AtK

+

for every

O(E)

K

(#K~),

+ o(E),

(16.13)

= A Y " - AcK0

where the order of magnitude of Ac, At's, Aq's is abbreviated as E. (Note that At's as well as Aq's tend to zero as Ac does by virtue of Theorem 16.3 and the smoothness of C.) Since A t is a loop and A q a cutset, (16.13) shows that the AtK'sand Aq,'s can be regarded, up to the first order of magnitude of E , as the flows and the tensions in the linear resistor network whose topological structure is the same as that of the given network, whose branches have conductances $""(OK), and in which only one current source is placed in parallel with bK0. Similar arguments apply also to the other three cases. Hence, the relations (16.10) follow immediately from the reciprocity properties of linear resistor networks expounded in Appendix 3. Q.E.D.

(p,

The assumptions regarding the smoothness of C,'S at q,), which we adopted in the preceding theorem, are essential, as the reciprocity does not hold for a network such as that shown in Fig. 16.3. (The reader should verify it.)

108

IV. Network-Flow Problems in General

Fig. 16.3. Failure of the reciprocity.

16.4. Existence

As was shown in Theorem 15.5, a necessary and sufficient condition for the existence of solutions for a network-flow problem is that there exist both a feasible flow configuration and a feasible tension configuration. Hence, it is important to investigate the conditions for the existence of feasible configurations. To begin with, we introduce new convex functions and cp', associated with a branch characteristic curve C, . As before, we denote the projection of C, to the 5 axis by I" = [b", cK] (b" may be -00 and/or cK may be 00 if I" is an infinite interval) and the projection to the q axis by J, = [d, , e,]. The function p is defined {on the whole real axis ( - 0 0 , a)when I" is finite, on (- co,01 when I" is semi-infinite to the right, on [0, 00) when I" is semi-infinite to the left, and on the point [0, 01 when I" is infinite in both directions} as p(q)= c"q for q > 0, = b"q

for q < 0,

=O

for q

(16.14)

= 0.

Similarly, the function q, is defined {on the whole real axis (- 00, 00) when J, is finite, on (- co,01 when J, is semi-infinite upwards, on [0, 00) when J, is semi-infinite downwards, and on the point [O,O] when J, is infinite in both directions} as E(t)= e,t for t > 0, = d,t

for

=O

for

t < 0, t = 0.

(1 6.15)

Furthermore, we write, for an arbitrary vector 6 [E X (R)] and an arbitrary vector q [E Y(R)], -

Y'(11)=

c

bxeB

V(V,),

W)=

c

b,ES

cp'dr").

(16.16)

109

16. General Properties of Solutions

It should be noted that 3" (or qK)may be regarded as one of the convex where functions associated (in the sense defined in $5.2) with C, (or cK), E , is the curve that consists of the horizontal segment coinciding with the axis on I" and of the vertical segments with abscissas b" and cK,and C is the curve that consists of the vertical segment coinciding with the r] axis on J, and of the horizontal segments with ordinates d, and eK(cf. Fig. 16.4). is then The other convex function @, (or p)associated with C, (or defined on I" (or J,) and has the value 0 everywhere on the domain.

c,)

5"

-c

(b)

(a)

Fig. 16.4. Modification of C, to

cKor

Theorem 16.5. A necessary and sufficient condition for the existence of a feasible flow configuration on a network N ( 6 ;C,, . .. , C,) is that -

'y(q) 2 0

(16.17)

for any cutset q [E Y(R)] in the domain of T.A necessary and sufficient condition for the existence of a feasible tension configuration is that (16.18) for any loop 6 [E X(R)] in the domain of 5. [Here, we shall regard (16.17) or (16.18) as being satisfied byanq or6 not in the domain of T or 5.1Moreover, a feasible configuration exists if (16.17) or (16.18) holds for every elementary cutset q or every elementary loop 6.

PROOF. We shall give a proof for a feasible flow configuration. Consider the network M ( 6 ; C,, . . . C,). It has evidently a feasible tension configuration, i.e., one in which the tension across every branch is y

IV. Network-Flow Problems in General

110

equal to 0, so that, by virtue of Theorem 15.5, a feasible flow configuration exists if and only if v(q)has a lower bound on its domain. Moreover, M ( 6 ; C,, . . . , C,) has a feasible flow configuration if and only if .,Y((C,; . . . , C,) has one, since C, and have the same I" in common. Since @,(t;) = 0 (( E l K )we , have

el,

e,

'(6)

(PK(5")

b, E S

=O

if a feasible flow configuration 5 exists. Hence, we must choose 0 as a lower bound of T(q),which proves (16.17) (Theorem 15. I). What remains to be proved is that, if (16.17) holds for every elementary cutset, then it holds for every cutset. To do this, we make use of the equisignum decomposition in Appendix 2. An arbitrary cutset q is decomposed into an equisignum sum: q = Y I q + Y z 9 + .. I

(16.19)

. 7

2

where y i > 0, the q's are elementary cutsets, and q 3 q. Since i

i

y], i

is either

null or has the same sign as q K , we have

so that we have (16.20)

Thus, we have proved that if (16.17) holds for every elementary cutset then Q.E.D. it holds for every cutset. 17. TWO-TERMINAL CHARACTERISTIC

I n $13.3, we investigated how the characteristic of a composite branch is related to the characteristics of component branches in the case that the composite branch is a series-parallel connection of component branches. I n this section, we shall consider the same problem for the general connections of branches. As is illustrated in Fig. 17.1, we consider a network with n - 1 branches, calling two specified nodes entrance and exit, respectively. At the entrance and the exit, and there only, flows can go into and out of the network from/to the outside. Our concern is the relation of the tension (i.e., potential difference) between the entrance and the exit to the flow from the outside. In order to investigate the relation, it is convenient to introduce an extra

111

17. Two-Terminal Characteristic

I

Internal branches b,, . . . , bn

I

I

---f

Reference branch b,

Fig. 17.1. Two-terminal network.

branch (to be named b,) that connects the exit node directly to the entrance. The extra branch will be called the reference branch and the other branches the internal branches. The characteristic of the “composite branch,” which is composed of the internal branches and which has the entrance and the exit as its two endnodes, can be measured by taking various kinds of branches as the reference branch, e.g., by taking a voltage or current source and changing its value parametrically. A network of this kind will be called a two-terminal network and be denoted, e.g., by N ( 6 ;ref., C , , . . . , C,), where 8 is the graph containing the reference branch as well as the internal branches, the CK’sare the characteristics of the internal branches, and “ ref.” means that the first branch is the reference branch, i.e., that its two endnodes are the entrance and the exit. Let us denote by 5 the total flow that enters the entrance, passes the internal branches, and goes out of the exit, and by q the difference of the potential at the exit from that at the entrance. 5 and v are regarded as the flow in and the tension across the composite branch, respectively. Then, the flow in the reference branch is equal to that in the composite branch, and the tension across the former branch is equal to the negative of that across the latter:

5=t’,

v=-q1.

(17.1)

Our problem is to obtain the set C of (5, q)’s, i.e., (<’, --q1)’s, for the twoterminal network with a given Q and given characteristics of internal

112

IV. Network-Flow Problems in General

branches. The C is regarded as the characteristic of the composite branch, which we shall call the two-terminal characteristic of the two-terminal network. The principal aim of this section is to show that the two-terminal characteristic of a two-terminal network satisfies properties (i)-(iii) in $5.2, so that a composite branch may be treated as if it were a single branch. First, let us assume that, if we take as the reference branch b, a linear resistor of unit resistance, then the two-terminal network has a solution of Problem 111, i.e., that the conditions (16.17) and (16.18) in Theorem 16.5 are satisfied by any cutset q and any loop 6 such that car q $ bl and car 5 b,. [Here it should be noted that, in this case, I' = 5, = (- co, co), and, consequently, (16.17) or (16.18) is regarded as being satisfied by any q or 5 such that car q 3 b, or car 5 3 b,.] For, otherwise, the C would be an empty set. Then it is readily seen that C intersects the line ((5, q ) I 5 + q = t } for any 1. In fact, if we take as the reference branch b, the series connection of a linear resistor of unit resistance and a voltage source of value - t , i.e., if we put C, = { ( P ,ql) I ql = t1- t } , we always have a solution of Problem I11 by virtue of the preceding assumption as well as the fact that I' =J1 = (- co, 00). Due to (17.1), the solution should satisfy 5 + q = t . Thus, property (ii) in $5.2 is proved to be satisfied by C , since property (i) in $5.2 (to be proved just below) prevents more than one point from lying on a line ((5, q ) 1 5 q = t } . Next,-let us suppose that there exist two points (5, q ) and (5, q) on C

+

1

2

1

2

such that

(5 - 5 ) h 1

2

1

< 0.

(17.2)

2

This means that, if we take as the reference branch b, a branch whose characteristic curve passes the two points ( < I , ql) = (5, - q ) and (t', ql) =

(5, - q ) (such a branch does exist), then 2 2 (6,q) and (5,q) of Problem I I1 such that 1

1

2

1

1

1

1

2

2

the network has two solutions

2

t1Z 5 ' 1

2

and

q1 # q l . 1

2

This contradicts the assertion of Theorem 16.2. Hence, C satisfies property (i) in $5.2. Properties (i) and (ii) assure the continuity of C. The remaining problem is to show the closedness of the projections of C to the 5 and the q axes. Suppose that there is a point (5, q) on C. The network has a solution of

113

17. Two-Terminal Characteristic

Problem I11 if we take as the reference branch b, a current source of value (. Conversely, if the network has a solution with a current source of value 5 as the reference branch, then there exists an q such that (5, q) E C [because J1 = (- 00, 00) in this case]. Hence, the projection Z of C to the 5 axis consists of those 5’s which satisfy n

(1 7.3)

c n

for an arbitrary cutset q

=

K=

q, b, (Theorem 16.5), where

1

V’S( K = 2,

. .., n) are the functions defined in connection with the characteristics (2,’s of the internal branches in $16, and it should be noted that for a current source of value 5, we have $‘(?I

=

t?.

(17.4)

Since (17.3) is homogeneous in the ?,Is, we have

c V(V,), where min is taken over all the elementary cutsets q c n

-min

1

C

$%,> 6 5 6 min

K=2

q,=-1

n

(17.5)

x=2

n

I], bK such that

=

,= 1

ql = 1 or - 1. (Note that there are a finite number of such cutsets.) (17.5) gives a necessary and sufficient condition for the 5 to belong to I. Therefore, the projection Z is a closed interval. Similarly, the projection J of C to the q axis is also a closed interval. Thus, we have

Theorem 27.2. The two-terminal characteristic of a two-terminal network of an arbitrary structure possesses all the properties (i), (ii), (iii) (in $5.2) which a branch characteristic should have. We shall postpone the development of practical algorithms for obtaining the two-terminal characteristics until Chapter V, and study a macroscopic relation between the tension and the flow of a two-terminal network such as follows.

Theorem 27.2. Let C be the two-terminal characteristic of a twoterminal network N ( 0 ;ref., C,, . . . , C,), let ((, ?) and ((, ?) be any two points on C, let the (t”,qK)’s and 1

the C,’s

( K = 2,

1

(t”,q,)’s 2

1 1

2 2

be the corresponding points on

2

...,n) [i.e., let (6,q) and (5,q) be the corresponding 1

1

2

2

114

IV. Network-Flow Problems in General

+ c {"b, n

6 = tb,

solutions of Problem 111, where

i

i

and q = -qbl i

~ = 2 i

i

+

n

1I], b,,, ~

let 43 and $ be the convex functions associated with C,

= i2

and, finally, let the q K ' s and $,'s be those associated with the CK's ( K = 2, ..., n). Then, we have

(17.6)

n

n

PROOF. We shall prove only the first equation, since the second can be proved similarly. First of all, we note that the right-hand side (of the first equation) of (1 7.6) does not depend on the choice of (t",I]")'s and (t",11")'s. 1

2

1

2

In fact, if we take as the reference branch a current source of value 5 i

( i = 1 or 2), then we have cp,(t) = 0, so that we have i n

(17.7) which, being the minimum value, is evidently independent of the choice of We have to prove that

(t",11,)'s. i

i

(17.8)

where the integrals are taken along the C and C,'s, respectively (cf. Remark 5.3 in $5.2). Since we may assume, without loss in generality, that 5 > 5, 2

we can choose, for an arbitrary positive number (5, ij), . . ., i j ) on C such that 0

(e,

E,

1

a sequence of points

N N

0

t = f g t s . * . s f s[=t,

At=

~ = i i s i j s . * N*ii- 1s s iNj = q2 ,

Aij= i j - i j < ~ .

1

1

0

0

N-1

1

1

N

2

i

i

5

i+l

i+l

-(<E, i

(17.9) i

The corresponding values F" of the flows in the internal branches as well as i

those ijK's of the tensions may not be monotone. However, as will be seen i

if we take as the reference branch a current source (voltage source) and

115

17. Two-Terminal Characteristic

change its value successively from

t to Nt ( iOj to Nij), we can choose them in 0

such a way that

- FI 5 A t ,

1AFI = I i

i+l

i

IAij,I

i

=

1 i j - ij,l

i

Aij

i

i+l

(17.10)

i

(cf. Theorem 16.3). Obviously, we have (17.1 1)

(cf. Fig. 17.2), and, similarly,

5 IAijKl 1AF1

E

*

i

At.

i

(17.12)

i

C

Fig. 17.2. Proof of (17.11). n

Furthermore, since the tensionconfigurationq i

difference of flow configurations Ag i

i

= Atb, + i

i.e.,

5 E At i

+ ( n - 1 ) A~t = ne A t , i

+ K = Z ij, b,

E - ijbl

i

i

and the

c A F b , are orthogonal, n

K=Z

i

116

IV. Network-Flow Problems in General

and, hence,

Since, in (17.13), n and

2

t are fixed, whereas E is arbitrary, we are led 1

to (17.8).

Q.E.D.

Theorem 17.2 affords means to calculate the increment of the sum of the cp,(t")'s of the internal branches by means of the two-terminal characteristic without knowing the flow configurations of the internal branches. 18. PROBLEMS OF APPROXIMATION

Practical algorithms for solution of network-flow problems are well developed, especially in the case where all the branch characteristics are step-functions (and, sometimes, piecewise linear functions). Therefore, we often have to solve a problem with general branch characteristics by approximating these characteristic curves by means of simpler curves. Or, it will often be the case that we are unable to know the exact forms of the cp,, functions or the C, curves when we formulate problems of the actual world in mathematical terms. In any case, it is important to investigate the manner of dependence of the solution upon the variations in branch characteristics. There are two ways to express the degree of approximation when a feasible configuration 6 (or q) does not necessarily coincide with an optimal (or lq, - f),I); the one f (or 4). One is to measure the differences 15" [or ul(q)] from a({) [or Y(fi)]. other is to estimate the difference of @(t) We begin with the latter, which is easier. $K

eKI

Theorem 18.1. Let 6 be a feasible flow configuration, let q be a feasible tension configuration, and let (f,4) be a solution of Problem 111, respectively, for a network M ( 6 ; C,, . . . , C,,). Furthermore, let cK denote the area enclosed by the characteristic curve C, of branch b, , the horizontal and the vertical straight line passing the point (t",q,) (cf. Fig. 18.1). Then we have n

c a".

1@(5) - @({>I+ IVq) - Y(4)I I

(18.1)

K = l

PROOF.

Since

@(k) - @(Q 2 0,

Y(q) - ~ ( f i 2 ) 0,

@(f)+ ~ ( 4 =) 0,

18. Problems of Approximation

117

Fig. 18.1. Definition of a x .

and

5 . 1 = 0, we have

C@W- @@I + CWS) - Wil)l - 5 - 1 = + V1) - 5 * 1 n

=

CVK(5")

K=

-k

$K(VK)

- 5"1K19

(18.2)

1

where each term in the brackets on the right-hand side of (18.2) is equal to Q.E.D. 0, [cf. $5.2, (5.46), and Fig. 5.21. Let (c, 4) be a solution of Problem 111 for N ( 6 ;C,, (c',4') be one for N ( 6 ;C1',.. . ,Cn').Then we have

Corollary 28.2.

.. . , C,) and let

where @ and Y are the objective functions for the former network, and (18.4)

(cf. Fig. 18.2).

Fig. 18.2. Definition of T

~ .

118

IV. Network-Flow Problems in General

Corollary 18.2. For a given branch characteristic C,, let C,' be a step-function approximating C, such that every point on C,' is at a distance, at most 6 horizontally and at most E vertically, from C, (cf. Fig. 18.3). Then the T,, defined in Corollary 18.1, does not exceed ES.

Fig. 18.3. Step-function approximation.

Theorem 18.1, as well as its corollaries, affords means to evaluate the difference of the value of the objective function for a nonoptimal feasible configuration from that for an optimal one. Next, we shall investigate methods of evaluating the difference of the value of the flow or tension in a branch corresponding to a nonoptimal configuration from that for an optimal one. Theorem 18.2. Let 6 and q be a feasible flow configuration and a feasible tension configuration of a network N o= N ( 6 ;C,, . . ., C,,),such that for each K , point (t",q,) is at a distance 6" horizontally and E, vertically from C,, i.e., that

6" = min

It'' -

5" E,

= min

[(g", q,)

E

C,], (18.5)

lVK

- i.1

C(5", ii,) E C,l.

rl"

Then there is a solution (&G) of Problem I11 such that, for each K,

(see Fig. 18.4).

PROOF. It suffices to give a proof for flows, since the proof for tensions is similar. We denote by CK'(6,)or C,'(-6") the curve obtained by the horizontal translation of C, by 6" to the right or to the left, respectively. Then either C,'(SK)or C,'( - 6') passes the point (F, q,). The modification

119

18. Problems of Approximation C,

Fig. 18.4. Horizontal and vertical distances from a point

(p,7.)

to C,.

of C, to C,'( 6,) corresponds to the attachment of a current source of value fa" in parallel with the branch b,. Furthermore, we denote by CK'(tK, q,) the curve which, satisfying properties (i), (ii), (iii) in $5.2, lies as near C, as possible (see Fig. 18.5), i.e., we set

{(t",

(t",

qK)l

f K )

' K ' ( q K

-

- VK) 2

qK)(qK

u {(F, qx)I3 (gK, VJE C K : ( t " - E")(e"- 8.)2 O>* (18-7) Set

5 = 5,

q =q,

6" =6"

1

1

1

and consider the network N l

C,,'(<"q,)]. , Obviously, 1

1

(6, q) is 1

1

Then, modify Jlrl into Jlr,'

E

(K

= 1 , ..., n),

= N [ S ; Cl'(+6'),

(18.8)

C2'(t2, q 2 ) , . ..,

1

1

1

a solution for N l (cf. Theorem 16.1).

N [ S ;C1, C2'(t2,q2), . . ., C,'(t", q,)] by

attaching a current source of value

fa'1

1

1

1

1

in parallel with branch b,.

Theorem 16.3 assures the existence of a solution (5,q) for such N l ' such 2 2

that

1AY1 = 15" - Yl S 6' 1

2

1

for every K .

Fig. 18.5. Modification of branch characteristics: -,

---, Ci(56K).

(18.9)

1

C,; --,

C,'(p, qK);

120

IV. Network-Flow Problems in General

Moreover, if we denote by 6" the horizontal distance from (t;", qK) to C, , 2

2

2

we have

#=o,

6"66" 2

2

Again, it is obvious that

(6, q) is 2

(K=2

,.", n).

(18.10)

1

a solution also for N2= N [ O ; C,,

2

Cz'(fd2), C3'(t3, q3), . ..,C,'(t;", q,,)]. Proceeding in a similar way, we 2

2

2

have a solution (

6,

i+l

2

2

q ) for Xi' = N [ O ; C,, .. . ,Ci, Ci+l(t;i+l,qi+,), i

i+l

. . ., Cn'(r, v,,)], which is a solution for J V ~ =+ "6; ~ C,, i

i

..., Ci,

i

1~t;"l= Ii +t;"l - t;"1 6 6' i i

for every K

( 18.11)

1

and 61 =

... =

i+ 1

6'

=o,

i+ 1

6" 5 6"(5 6" 5 i+l

i

5 6")

(K = i

+ 1 , . . . , n),

(18.12)

i-1

6" denoting the horizontal distance from (t",q,) to C, . Thus, we finally get i

a solution (

6 ,q

i

i

) for Nn'= N ( 6 ;Cl, . . ., C,) = N o ,i.e., we get

n+l n+l

(18.13)

where

n

n

n

19. EQUIVALENCE TRANSFORMATIONS OF NETWORKS

It sometimes happens that even if we modify the topological structure or the branch characteristics of a part of a network in a certain manner, the solution continues to have the same values of the flows in and the tensions across the branches of some part of the network. The modifications of this sort are called equivalence transformations.

19. Equivalence Transformations of Networks

121

We shall list the typical equivalence transformations in the following paragraphs : Type I: bisomorphism. If 8, and 8 , are mutually 2-isomorphic graphs (each with n branches), then the continuity conditions for flows and tensions have the same form in N , = N ( 6 , ; C1, . . . , C,) as in N , = N ( 6 , ; C1’, . . ., C,’), as is evident from the definition of 2-isomorphism (Chapter 111, $11.3). If, furthermore, the characteristics C, and C,’ of the corresponding branches are the same in N l and N,,

c, = C,’

(K

= 1, . . ., n),

then a solution for MI is obviously a solution for A’-,, and vice versa. As was shown in $11.3, a nontrivial 2-isomorphic transformation consists in reversing a two-terminal subgraph followed by reorienting all the branches contained in it (see Fig. 19.1).

Fig. 19.1. Type 1 transformation.

The type-1 transformations that we encounter most frequently are the reordering of series connection of branches or two-terminal subgraphs, as is illustrated in Fig. 19.2, where (b) is obtained from (a) by first reversing the two-terminal subgraph consisting of all the two blocks and then reversing each of the two.

(a)

(b)

Fig. 19.2. Reordering of series.

122

IV. Network-Flow Problems in General

Type 2: Transformations concerning special branch characteristics. The transformation mentioned in Theorem 16.1 can be regarded as a most general one of this type, but we shall be concerned with more special ones. Type 2.1: Transformations concerning current sources. As is obvious from the continuity condition for flows expressed in the form (13.1), (13.2), or (1 3.3) and from the characteristic of a current source, if we modify the arrangement of current sources in such a manner that the sum of their values in an arbitrary elementary segregation (the signs of branches being taken into account) may not vary, the solution remains invariant for the remaining branches. For example: (1) We can construct a network ."equivalent to a given Jf by opencircuiting all the current sources of Jlr and then connecting, in parallel with each branch of a tree on the resulting network, a current source of the appropriate value. (2) Replacing a current source of value c by the parallel connection of two or more current sources the sum of whose values is equal to c (or vice versa) does not affect the solution for the remaining branches (Fig. 19.3).

Fig. 19.3. Parallel splitting of a current source.

(3) Inserting, in series with a current source of value c, a branch whose characteristic curve contains a point with abscissa c (or vice versa) does not affect the solution for the remaining branches (Fig. 19.4).

Fig. 19.4. Series splitting of a current source.

(4) If the sum of the values of the current sources incident to a node is equal to 0, splitting them from the node (or vice versa) does not affect the solution (Fig. 19.5). Furthermore, after splitting, one of the current sources may be short-circuited without affecting the solution.

123

19. Equivalence Transformations of Networks

Fig. 19.5. Splitting a node.

(5) In general, let us denote by ca the sum of the values of the current sources incident to the node nu ,where we have necessarily

c

ca = 0.

a

If we open-circuit all the current sources of the given network and add a new node (which we shall denote by no) as well as the current sources, respectively, of value cayconnecting no to na (see Fig. 19.6), then we have a network equivalent to the given one in regard to the remaining branches. This kind of transformation may be derived as follows: first, we split each of the current sources of the given network into the series connection of two current sources of the same value according to (3), combine the middle points into one node no according to (4), and then replace parallel current sources by one according to (2). A network obtained by this kind of transformation may be regarded as a standard-form network, into which are transformed all the networks connected by the type-2.1 transformations "0

3

Fig. 19.6. Standard-form network.

124

IV. Network-Flow Problems in General

Type 2.2: Transformations concerning voltage sources. As is obvious from the continuity condition for tensions expressed in the form (13.4), ( 1 3 3 , or (13.6) and from the characteristic of a voltage source, if we modify the arrangement of voltage sources in such a way that the sum of their values in an arbitrary elementary closed path may not vary, the solution remains invariant for the remaining branches. For example : (1) We can construct a network N ’ equivalent to a given N by shortcircuiting all the voltage sources of N and then inserting, in series with each branch of a cotree on the resulting network, a voltage source of the appropriate value. (2) Replacing a voltage source of value e by the series connection of two or more voltage sources the sum of whose values is equal to e (or vice versa) does not affect the solution for the remaining branches (Fig. 19.7).

Fig. 19.7. Series splitting of a voltage source.

(3) Connecting, in parallel with a voltage source of value e, a branch whose characteristic curve contains a point with ordinate e (or vice versa) does not affect the solution for the remaining branches (Fig. 19.8).

fi--<-Fig. 19.8. Parallel splitting of a voltage source.

(4) If the sum of the values of the voltage sources constituting a path is equal to 0 so that the potentials at the two endnodes of the path are to be equal to each other, then the identification of the two nodes (i.e., combining them into one node) does not affect the solution (Fig. 19.9). Furthermore,

Fig. 19.9. Equipotential nodes with voltage sources.

19. Equivalence Transformations of Networks

125

after identification, one of the voltage sources may be open-circuited without affecting the solution. ( 5 ) If a current source is incident to one of the endnodes of a voltage source, we may reconnect the current source to the otherendnode of the voltage source without affecting the solution, as is shown in Fig. 19.10.

Fig. 19.10. Sliding a current source across a voltage source.

Type 2.3: Transformations concerning diodes. If the potential la at node n, is guaranteed by some reason to be greater than or equal to the potential lBat n B , we may add a diode connecting nB to na to obtain an equivalent network, and vice versa (Fig. 19.11).

Fig. 19.11. Redundant diode in parallel.

126

IV. Network-Flow Problems in General

If the flow in a branch b, is guaranteed by some reason to be nonnegative, we may insert a diode in series with b, to obtain an equivalent network, and vice versa (Fig. 19.12).

3-o-+--b,

c;" 2 0

Fig. 19.12. Redundant diode in series.

In the following, we shall give some concrete examples of the transformations of this type. (1) If an elementary segregation consists of diodes all placed in the same direction, then we may open-circuit all of them (Fig. 19.13). In fact, the flows in them must be nonnegative, and the continuity condition requires the sum of the flows to vanish so that the flow in each diode should vanish.

E-ED Fig. 19.13. Segregation by diodes.

(2) If an elementary segregation consists of diodes of which one is placed in the direction opposite to the others, then we may short-circuit the opposite diode (Fig. 19.14). In fact, the flow in the opposite diode would be constrained to be nonnegative owing to the presence of the other diodes if the opposite diode were short-circuited.

Fig. 19.14. Short-circuiting a diode.

19. Equivalence Transformations of Networks

127

(3) If an elementary closed path consists of diodes all placed in the same direction, we may short-circuit all of them to shrink the path into a single node (Fig. 19.15). In fact, the tensions across them must be nonpositive, and their sum must vanish, which means that the tension across each diode should vanish.

Fig. 19.15. Equipotential nodes with diodes.

(4) If an elementary closed path consists of diodes of which one is placed in the direction opposite to the others, then we may open-circuit the opposite diode (Fig. 19.16). In fact, the tension across the opposite diode would be constrained to be nonpositive owing to the presence of the other diodes if the opposite diode were open-circuited.

Fig. 19.16. Open-circuiting a diode.

(5) If there are only diodes connected to a node no, then we may remove no as well as the diodes connected to it and add new diodes as follows to obtain an equivalent network. Let us denote by n, , np , ny, etc., the endnodes of the diodes connected to no. We add a new diode connecting n, to nB if and only if there is a diode connecting n. to no and one connecting no to np (see Fig. 19.17). This kind of transformation is sometimes called a star-to-mesh transformation of a diode network, and is always possible. The reverse transformation, i.e., mesh-to-star transformation, is

128

IV. Network-Flow Problems in General

(a)

(b)

Fig. 19.17. Star-mesh transformation.

not always possible. (The reader should verify that, if the number of diodes connected to nodoes not exceed 3, a mesh-to-star transformation is possible.) We have given and shall give a number of examples of composing a branch with the same characteristic from branches with simpler characteristics in many different ways (e.g., in 913.3, 420.1, 921.1, etc.). It will be a good exercise for the reader to try to modify one way of composition into another by means of the transformations listed thus far.

V Linear Network-Flow Problems

By a “ linear network-flow problem ” we mean a network-flow problem in which all the branch characteristic curves are step-functions because such a problem is formulated as a “linear programming problem” from the viewpoint of mathematical programming. However, linear networkflow problems form a very special class of linear programming problems, so that we have efficient methods to treat them with, methods that take account of the special characters of the topological structures of networks. These methods are much more efficient for linear network-flow problems than the standard simplex methods. This chapter is devoted to the exposition of the algorithmic methods of solving linear network-flow problems as well as their applications to practical and mathematical problems. We shall begin with the simple problems that are concerned with flows or tensions only ($20 and $21), apply the results for them to the treatment of the two-terminal network problems (§23), and then turn to the general problems ( $24) where the approximate solutions of nonlinear network-flow problems will also be dealt with. As the reader will soon be aware, the general attitude of our expounding materials in this chapter is quite different from that in the preceding chapter. We have so far endeavored to avoid making use of the concept of “nodes” in order to endow the theory with the dual structure as much as possible, and we believe that we have succeeded in the dualistic 129

I30

V. Linear Network-Flow Problems

exposition of everything essential in network-flow theory. In this chapter, however, we are forced to make full use of “nodes” because it is practically better and more efficient to do so when solving a problem algorithmically. Thus, we shall sometimes give to a pair of problems that are theoretically dual to each other the solution algorithms that are not apparently dual. Nevertheless, the duality will be seen to creep into the essence of the solution algorithms in most cases. Unless otherwise stated, we shall assume throughout this chapter that the graph of a network (including the reference branch in the case of a two-terminal network) is nonseparable. 20. MAXIMUM FLOW

20.1. Presentation of the Problem Formally, the problem is to obtain the projection to the 5 axis of the two-terminal characteristic curve of a two-terminal network A’-(@; ref., C , , . . . , C,) for which each C, (K = 2, .. . ,n) is a characteristic curve such as shown in Fig. 20.1. I n a more natural expression, the problem is to obtain both the maximum and the minimum values for 5’ (the flow in the reference branch, i.e., the flow from the entrance node through the twoterminal network to the exit node) when the flows 5“ in the internal

qI b“

(d)

(e)

Fig. 20.1. Capacitated branch.

F t K

cK

20. Maximum Flow

131

branches (K = 2, . ..,n) are subject to the constraint relations r" E I" or b" 5 5 c" (I" = [bK,c"], C" 2 b"). (20.1) In the foregoing, it is understood that we put C" = 00 and/or b" = -a when there is no upper and/or lower bound for 5". As in 917, we shall denote the entrance and the exit nodes by ns and nt , respectively. (Sometimes, the entrance node is called the source and the exit node the sink; see Fig. 20.2.)

<"

Internal branches b, , ..., b,

y

,3

132

V. Linear Network-Flow Problems

QbP

Fig. 20.3. Transforming node capacities into branch capacities.

and the exit, i.e., by reorienting the reference branch b,.) The necessary and sufficient condition for the existence of a feasible flow configuration was given in Theorem 16.5. Since the number of elementary cutsets is finite, we can write out the relations (16.17) for all the elementary cutsets q's. Let us assume that (16.17) holds for every elementary cutset q such that car q $ b, ;for, otherwise, we can have no feasible flow configuration, whatever value c1 may take. Then, we have to consider the case where q is an elementary cutset with q, # 0, i.e., with ql = 1 or q, = -1. In such a case, (16.17) is written as

(-b'=)-

c1

p(qK)

s-

or

p(qK)

(c'=)b' 5

(20.2)

K=2

K=2

n

for any elementary cutset q

=

C q, b,

K=

with ql

=

1 or - 1, respectively,

1

where p(q)is the function defined in (16.14) and it is taken into account that 1 if q1 = 1, iwll)= C'Vl = c (20.3) $'(ql) = b'ql = -b' if ql = - 1, and bl = cl. (20.2) can be summarized into

to give the maximum for cl. As is easily seen, the elementary segregation corresponding to an elementary cutset with q, = - 1 contains in the positive (negative) direction to those of 'ill2 (?Ill), where the branches connecting the nodes of '32' (a2) (Wl, W2) is a dissection of the set of nodes W such that ns E a,and n, E !TI2.

133

20. Maximum Flow

We shall call the set of internal (i.e., nonreference) branches belonging to such a segregation (the direction of the branches being taken into account) a cut separating the entrance node from the exit node, and K=2

p(q,)= the sum of cK's such that

q, = 1

(20.5)

+the sum of (- bK)'ssuch that q, = - 1

its value, where qK= 1, - 1, or 0, according as branch b, is contained in the cut in the positive direction, in the negative direction, or is not contained in it, respectively. Thus we have from (20.4) the famous theorem due to Ford and Fulkerson [l]: Theorem 20.1 (Maximum-flow minimum-cut theorem). Through a twoterminal network of which each internal branch b, allows a flow between b" and cK, the maximum possible value of the total flow which can be carried from the entrance to the exit is equal to the minimum of the values of the cuts separating the entrance from the exit.

Figure 20.4 is an illustration of this theorem.

Segregation ~

I

I

I

\

\

<'

n3

Ref.

I I-

1

-1

-

Value

cut 1

0

I I

b,

1 0 1 - 1

0 1

0 0

1 0 1 0 1 1 O 0 0 1 I b, b, b4 b, bb

8 6

7 8

(Value of the minimum cut = 6)

n3

(Maximum flow

= 6; ft:saturated

branches)

Fig. 20.4. Illustration of the maximum-flow minimum-cut theorem: Z2 = [0,6],

Z3= [0,2], Z4 = [-3,4], Z5 = [0,1], Z6 = [0,7].

134

V. Linear Network-Flow Problems

REMARK20.1. If every internal branch has nonnegative capacities, i.e., if L'. 2 0 and - 6" 2 0 for all K = 2, . . . , n, then P(q)is always nonnegative :

P(q)= c"q 2 0 =

b"q 2 0

for q

>= 0

for q 5 0

(20.6)

so that (16.17) always holds for a cutset whose carrier does not contain b,. It should also be noted that the validity of (16.17) for cutsets whose carriers do not contain b, is equivalent to the existence of a feasible flow configuration in the network N ' obtained from the original network by shortcircuiting b, .

REMARK 20.2. Theorem 20.1 was proved on the basis of Theorem 16.5. It can be proved directly as a by-product of the solution algorithm to be developed in $20.3.1. Furthermore, Theorem 16.5 can be proved on the basis of Theorem 20.1 by means of an equivalence transformation such as that which will be utilized in $20.3.2 to establish a solution algorithm for the general case. 20.3. Solution Algorithms 20.3.1 ALGORITHM FOR

THE SPECIAL CASE

An internal branch b, is said to be saturated in the positive direction if the flow 5" in it is equal to C" (the capacity in the positive direction), saturated in the negative direction if - 5" is equal to - b" (the capacity in the negative direction), and unsaturated otherwise, i.e., if b" < 5" < c"; or, simply, it is said to be in state S', S - , and So, respectively (cf. Fig. 20.5).

Fig. 20.5. Saturation states of a branch.

135

20. Maximium Flow

When the flow in branch b, is

tK,there

is a positive At" such that

5" + AtKsatisfies the capacity restriction if and only if t" < cK,i.e., if and

only if b, is not in S'. Similarly, there is a negative AtKsuch that 5" + AtK satisfies the capacity restriction if and only if b, is not in S - , and there is such a positive At" as well as a negative At" if and only if b, is in So. In the respective cases, we say that b, is conductive in the positive direction, conductive in the negative direction, and conductive in both directions with regard to incremental flows, and we call C" - l",or 5" - b", the residual capacity in the positive, or negative, direction. In this section, we shall give an algorithm for constructing a flow configuration that maximizes the total entrance-to-exit flow for the case where cK 2 0 2 b" for all the internal branches. First of all, let us note that the flow configuration 6 = 0 satisfies all the 0

capacity restrictions. Next, let us assume that we have a flow configuration

5= i

C t"b,

satisfying the capacity restrictions

~ =i l

5" 5 C"

b"

(K

= 2,

. ..,n)

(20.7)

i

with the total entrance-to-exit flow

tl. If there exists a

conductive path

i

Pi from the entrance to the exit, i.e., a path from the entrance to the exit such that every branch on Pi is conductive with regard to incremental flows in the direction in which it lies on Pi, then we construct a flow configuration 5 =5+N (20.8) i

i+l

i

where, for K # 1, A ~ K =

e

if

b, lies on Pi in the positive direction,

-8

if

b, lies on Pi in the negative direction,

i

=

=O

(20.9)

if b, does not lie on Pi,

and 8 is the minimum of the residual capacities (in the positive or negative direction according as the branch lies on Pi in the positive or negative direction) of the branches on Pi (so that 8 > 0). Obviously, we should have

t1= r1 + 8 > p,

i+l

i

(20.10)

i

i.e., the total entrance-to-exit flow is increased by 8, and

6 satisfies all i+l

the capacity restrictions.

136

V. Linear Network-Flow Problems

Roughly speaking, the algorithm for attaining a maximum flow consists in repeating the foregoing process. In the process of determining 6 from

6, the

if1

most essential step is to find a conductive path P i from the

i

entrance to the exit. This problem is a kind of time-honored “labyrinth problem,” for which the following algorithm would be computationally most efficient.

Labyrinth Algorithm 1. To each node na we assign a pair of signed numbers (kA, , +pa). They are determined in accordance with the rules described below, and, at the final stage, pa (or Au) is the node number (or the branch number) of the node (or branch) adjacent to the node nu along a conductive path from the entrance to node n, so long as such a path exists. (i) Initialization: To begin with, we assign (+ 00, -0) to the entrance node n, and (+ co, co) to all the other nodes n, (a # s). (ii) We take up a node n, whose pa has the minus sign, convert the minus sign before the pa into the plus sign, and perform the following operations for each branch b, in 8+n, v 6-n,: (iia) If b, E 6+n, is conductive in the positive direction and if the pair (+ co, co) is assigned to the node no. = 8-b,, then we convert the pair for n,, into (+ K, -a); otherwise we do nothing. (iib) If b, E 6-n, is conductive in the negative direction and if the pair (+ 00, +co) is assigned to the node n,. E P b , , then we convert the pair for n,. into ( -IC, -a); otherwise we do nothing. (iii) We repeat (ii) until we cannot decrease any longer the number of nodes that have the pair (+ 00, co). If p, # 00 for the exit n t , there exists a conductive path P from the entrance n, to the exit n, . In fact, we put a,, = t and determine ai’s, ti's, and K ~ ’ Ssuccessively by

+

+

+

q + 1

=pa,,

Ei+lKi+l=+IZ-,i

( i = o , l ,...)

(20.11)

from the pairs of (+A,, +p,)’s thus obtained. Due to the manner of determining the pairs, we should have a, = s for some r. Then the sequence of nodes (n,s = n a r , na,-l, . .., nal , nao = n,) and the sequence of signed branches ( E , bKr, .. . ,c1 bKl)define a required conductive path P. If p, = 00, we assert the nonexistence of a required path. For, if one existed which had the node sequence (n, = na,, . .., nao = n,) and the branch sequence ( E , bKr,. . . , ElbKI), then we should have pa,+, # co and pa, = co for some i (in fact, pa, = 0 and /la,= a);then, because of the conductivity of the path, b,, should be conductive in the positive or negative direction

137

20. Maximum Flow

(according as ei = 1 or - 1); in such a case, however, operation (ii) could be performed to convert Bai into a finite number, i.e., into c ( ~ + ~ .

EXAMPLE.Let us solve the maximum-flow problem for the network of Fig. 20.4. We set

60 :

(51; 5 2 , 0

0

...,

56) 0

= (0; 0, 0,0, 0,O).

The residual capacities (which are the same as the capacities) are as tabulated in (20.12): K

%-b"

Cr-% 0

2

6

3 4 5 6

2

0

4 1

7

0 0 3 0 0

(20.12)

The process of searching for a conductive path Po according to Labyrinth Algorithm 1 is performed as shown in (20.13), where we rewrite all the pairs of (*Aa, &fia)'s [except for the pairs ( + m , +a)]each time an operation (ii) is performed and an arrow indicates the node for which we perform operation (ii) :

(20.13)

Since fP4 = - 2 # co,we have a conductive path P o , which is determined as follows: go

= t = 4,

a1 = p4 = 2, u2

= p2 = 1 = s,

Po: ( n s = n 1 , " 2 , n 4 = n , ) ,

EIKl

=

&A4 = + 5 ,

E2K2

=

&A2

(+b2,

=

+b,).

+2;

138

V. Linear Network-Flow Problems

The incremental flow is assigned to b, and b, along P o , whose amount is equal to

8 = min(c2 -

c2, c5 - r5) = min(6, 1) = 1.

0

0

Thus we have 5=5+A5=(1;1,0,0,1,0), 1

0

0

where At' = 8 = 1 is the increment of the total flow. 0

The residual capacities for

5 and

the process of searching for a con-

1

ductive path are shown in (20.14)-(20.16) : P-b"

cK-%

K

1

2

5 2 4

3 4 5 6

I

s=l s=l 2 3

= 4,

1 0 3 1 0

0

a

a0

1

(A, -+m (+a, -0)c +O) (+2, - l)+ (+3, -1)

(+a),

s=l 2 3

(+a,+O) (+2, + l ) (+3, -l)+

s=l 2 3 r=4

(+a,+O) (+2, +1) (+3, +1) (+6, -3)

a, = p4 = 3, a 2 = p 3 = 1 = s, p1:

(20.14)

(n1, n 3 , n4),

E

(20.15)

~

E~ K~

(+ b3,

= K

&A4 ~

=

+6,

= +A3 = +3;

+ b6).

(20.16)

20. Maximum Flow

139

The amount of the incremental flow to be assigned to b, and b, along PI is

8 = min(c3 -

c3, c6 - 5,) 1

= min(2, 7) = 2,

1

and we have

6 = 6 + A6= 2

1

(3; 1,2,0, 1,2).

1

Proceeding in a similar manner, we have the residual capacities (20.17) for 6 and a conductive path P2 shown in (20.19) by means of the labyrinth 2

search (20.18) : K

$-b"

c"-$ 2

2

5

3 4 5 6

0 4 0

5

2

1 2 3 1 2

(20.17)

(20.18)

140

V. Linear Network-Flow Problems

The incremental flow is assigned to b, , - b, , b, , whose amount is 6 = min(c2 - t2,t4 - b4, c6 - 5,) 2

so that we have

E,

= E,

3

2

= min(5,3,

5) = 3,

2

2

+ A& = ( 6 ; 4,2, -3,

1, 5).

2

The residual capacities for

5 is as shown in (20.20) and the labyrinth 3

search algorithm proceeds as shown in (20.21): K

c"-% 3

2 3 4 5 6

5"-b" 3

4 2 0

2 0

7 0

1

2

5

(20.20)

(20.21)

Since we have j?, = p4 = co (not explicitly shown) in (20.21), we assert that there is no conductive path, so that E, gives the maximum total entrance-to3

exit flow t1 = 6. The cut of the minimum value consists of the branches that 3

connect node n, or n2 with finite j?, to n3 or n4 with infinite 8,. We have two things to prove. We must first show that the flow configuration obtained by means of the foregoing algorithm indeed gives the maximum value of the total entrance-to-exit flow, and second, we must prove that we never fail to attain a flow configuration giving the maximum total flow, or to know the possibility of the total flow becoming infinitely large, after afinite number of operations. Let us begin with the firstproblem. Let be a feasible flow configuration for which no conductive path can be found from the entrance to the exit, and suppose that there were If we another feasible flow configuration [ that had 5' greater than

p'.

141

20. Maximum Flow

t

consider the equisignum decomposition (see Appendix 2) of A6 = - f into minimal loops,

~6 =

- { = ~4

-+ ~6 -I-

1

then some Ays, say

At,

should have a positive At'. There exists an ele-

1

i

,

2

1

mentary loop of which the minimal loop A6 is a multiple. The branches 1

of the elementary closed path corresponding to the elementary loop, excepting the reference branch, form a path from the entrance to the exit. Since the AtK'shave the same sign for a fixed K and i

bKsgKgcK, b = 5

K < * K <

for all K ( # I ) ,

= cK

we have

bK-

6 At" 5 cK -

for all K ( # 1).

(20.22)

1

The AtK'sfor the branches on the path are nonnull, and, moreover, the 1

A Y for a branch lying on the path in the positive (or negative) direction 1

is positive (or negative), which means that the path is conductive with regard to f. However, this contradicts the assumption. Therefore, { gives the maximum p'. The rest of the problems to be investigated concerns the " finiteness " of the algorithm. In fact, if we permitted ourselves to perform the search for the conductive path arbitrarily, we might have an infinite sequence of flow configurations 6,6,5, ,. . such that 0 51 0

1

2

< 5' < * * * 1

and

lim

5'

i-m

i

=finite or

00,

according to the algorithm expounded above. In geometrical terms, if we consider a space with r K ' s as coordinates regarding a vector 6 as a point in it, then the set of loops or flow configurations 5's forms a linear subspace, and the set of feasible flow configurations forms a convex polyhedron in the subspace. There the locus of the sequence of 6's which has the i

undesirable property such as the above will appear as a zigzag curve exemplified in Fig. 20.6. Thus, we sometimes call a phenomenon of this kind a zigzag phenomenon. An exquisite example of zigzag phenomena with a finite lim t' was devised by L. R. Ford, Jr. (see Ford and Fulkerson i-rm i

[l]). Here we shall give a simpler example with an infinite lim 5'. i+m

i

142

V. Linear Network-Flow Problems

(b)

(a)

Fig. 20.6. Zigzag phenomena.

EXAMPLE.Let us consider the maximum flow problem on the network of Fig. 20.7, through which we can evidently assign as large a flow as we want from the entrance to the exit. However, what will result if we perform the flow augmentation in the following way? Let us denote by P, and P2the paths from the entrance to the exit defined by means of the branch sequences in (20.23) : (20.23) Starting from 6 = 0, we see that P, is conductive for 6 and that the maxi0

0

mum amount of incremental flow along it is 0 = 1, and we get

5 = ( l ; 1,0,1,0,1,0,1,0,1). 1

b, is brought into state S'. Then we find that P, is conductive for

6 so 1

that we get

5 =(2;

1,1,1,1,0,1,1,1,1).

2

Fig. 20.7. A network. l2 = I'

=

I4 = I5 = I'

= 1' = 1' = 1'' = [0, a); l6= [0, 11.

143

20. Maximum Flow

b, is brought into state S - . Again, P, is found to be conductive for

6. In 2

this way, P, and P2 become conductive alternately, and we have, in general,

6 = ( 2 i ; i , i , i , i , 0, i , i , i , i ) , 2i

6 = ( 2 i + l ; i + l , i,i+l, iyl,i , i + l i , i + l ) 2i+ 1

(see Fig. 20.Q with 5' = i. Thus, the amount of total flow increases by i

one in each step so that we have to continue the solution process without end.

5 - 2 i5- I

Zi+ I

2i

Fig. 20.8. Two paths.

We can get rid of zigzag phenomena if we observe the following antizigzag rule in searching for a conductive path. Antizigzag Rule. One must adopt as a conductive path along which an incremental flow is assigned a path consisting of unsaturated branches, i.e., branches in state So, so long as such a path exists. One may adopt a path containing branches in state S + or S - only when there can be found no path consisting of only unsaturated branches. The antizigzag rule states that in order to find a conductive path, we have to first apply the labyrinth algorithm to unsaturated branches alone, and if we fail to find such a path we then have to apply the algorithm to the entire network. The finiteness of the solution algorithm can now be proved as follows. Let us call the sequence of n - 1 symbols

m)=

(S2 Y

* * * Y

S,)

the saturation pattern of the internal branches with regard to the flow configuration 6,where S, is either S', S - , or So, indicating that branch b, is in the respective state. According to the antizigzag rule, each time

144

V. Linear Network-Flow Problems

we find a conductive path of unsaturated branches for E,, we either get a i

new flow configuration E, such that at least one So in S(E,)is changed into i+ 1

I

S + or S - in S( E, ), or we find that we can assign as large an incremental flow i+ 1

as we want along the path. Therefore, after a finite number, say i, , of steps starting from a 5, we shall get a solution 5 = E, , or find a conductive path ii

0

along which an infinite flow can be assigned, or encounter the situation where we can no longer find any conductive path consisting of unsaturated branches. In the first two cases, we have no problem. In the last case, it is proved that any other feasible configuration E, with the same saturation pattern S(E,)= S(5) as 6 has the same amount of total flow 5' = 5, as

5 . In

ii

it

ii

fact, since S(5) = S(Q, 5" - 5" can be different from zero only in

ii

ii

ii

unsaturated branches. Furthermore, since there is no conductive path of unsaturated branches, there is no closed path consisting of the reference branch and unsaturated branches, so that 5 - E, should be a loop whose il

carrier consists of internal branches alone, i.e., we have 5' going from 5 to ii

5

=

tl. On ii

we search for a conductive path in the entire network.

ii+l

t

And, after a finite number of steps, say i, - i,, we shall get a solution = 5 , iz

or find a conductive path along which an infinite flow is possible, or encounter the situation where we can find no conductive path of unsaturated branches. In a similar manner, unless we have a solution or an infinite flow, we have a sequence of flow configurations (20.24) such that (20.25) Because of (20.25), among the saturation patterns S(Q, S(g), ii

. . ., no two

iz

can be identical to each other. However, the number of possible saturation patterns is evidently finite, so that the sequence (20.24) cannot continue endlessly, which means that we do not fail to get a solution or to know the possibility of an infinite total flow. This was to be demonstrated. (The reader should try to solve the problem of Fig. 20.7 using the antizigzag rule and to ascertain that the solution process terminates with finite steps in whichever path we may adopt at each stage.)

145

20. Maximum Ftow

REMARK 20.3. Instead of calculating the amount 8 of incremental flow after we find a conductive path, we may do the necessary calculations in performing the labyrinth algorithm. To do so, we make use of the triple ( fA,, fpa, 0,) in place of ( I,, f flu) in Labyrinth Algorithm 1. In the initialization operation (i), we put (fl,, &pS, 0,) = (+ co, -0, co) for a = s and (+~,,+~,,0,)=(+00,+00,0) for the other a's. If we +a)to ( + K , -a) [or(-rc, -a)] convert a ( + L a , . , +pup) from (+a, in the iteration operation (iia) [or (iib)], then we determine 0,. by

+

0,.

= min(8,,

cK - r")

[or

= min(0,,

tK - b K ) ] .

It will easily be seen that in this way, we shall have

e = 8,. Whether this modification of algorithm contributes to the improvement of computational labor or not will depend on circumstances.

REMARK 20.4. It is worth noting that the (f A,, fp,)'s obtained by means of Labyrinth Algorithm 1 not only afford information about a conductive path from the entrance to the exit but also contain the information about the conductive paths from the entrance to all the nodes in the network. If we need only a conductive path to the exit, we may stop the algorithm as soon as we get p, # 00 for the exit. In this connection, it should be noted that we did not need to examine or perform the conversion of (+&, &&.) for all the nodes in aTS'n, in (ii) of Labyrinth Algorithm 1. What we need to do is to make a conductive path from the entrance longer toward the exit; Labyrinth Algorithm 2 follows this line of thought. It is not clear to the author-and it would perhaps depend on circumstances-which of Labyrinth Algorithms 1 and 2 is computationally more efficient. Labyrinth Algorithm 2. To each nu we assign a number cr, and, moreover, we prepare M pairs ( f ai , mi) of numbers (one of which has a sign), where M is equal to the number of nodes. These numbers are determined in accordance with the rules described below, and, at the final stage, cli indicates the node number of the ith node along the required conductive path and m iindicates that the ith branch along the path is the with among those branches incident to the node nUi . cr, is used to remember whether node nu has ever been examined (a, = 1) or not (cr, = 0). We denote by pa the number of branches incident to nu and assume that the pn branches are arranged in an appropriate (i.e., arbitrary but fixed) order.

146

V. Linear Network-Flow Problems

(i) Initialization: T o begin with we set (fa,,ol)= (+s, 0), ( f a i , oi) = ( + co,0) for i = 2 , . . . , M , as= 1, and au = 0 for a # s. = .-.= aM = co. (ii) Let i be the number such that a, # co and (Such a number i is uniquely determined.) Among the ( w i + l)th, the (mi+ 2)th, . . ., the (psi - I)th, and the p,,th branches in 6+nai u 6 - n u i , we search for the first branch b, such that it is conductive in the positive direction, b, E S + n a i and a, = 0 where nu = a- b, or that it is conductive in the negative direction, b, E 6-n,, and a, = 0 where nu = a'b,. If we cannot find such a branch, then we set( fai,mi)= (+ co,0)andreturn to the beginning of (ii). If we find one and if it is the ( w i + v)th in d'n,, u 6 - n u i , then we replace the value of wi by w i + v, set (+a,+,, mi+,)= (+a,0) or ( - a , 0) according as b, E 6+nUi(i.e., nu =d-b,) or b, E 6-n,, (i.e., nu = d'b,), a, = 1, and return to the beginning of (ii). (iii) Repeating (ii), we finally have a, = t for some i = r, or we have (kc(,,w ) = (+ 00, 0). I n the former case, we get the conductive path with the node sequence

(nu1 = ns naz 9

9

*

f

*

9

nor =

n,)

and the branch sequence ('1

bKl 9

..

' 9

',-

1'Kr-

1)

where bKi is the with branch of 6+naiu 6-n,,, and c i is equal to the sign before ai+,.In the Latter case, we assert the nonexistence of a conductive path. Furthermore, in the latter case, it is seen without difficulty from the manner of determining the ou's that there is a conductive path from the entrance node ns to node nu if and only if nu = 1. REMARK 20.5. The computational labor needed for solving a labyrinth problem by means of Labyrinth Algorithm 1 or 2 is at most proportional to the number of branches. This fact will be verified if we note that the conductivity of a branch is examined at most twice by either of these algorithms. it is obvious that, once a conductive path is found, the computational labor to augment flow along it is proportional to its length (which is always smaller than the total number of nodes). REMARK 20.6. As a by-product of the solution algorithm developed in this section, a direct proof of Theorem 20.1 is obtained. Suppose that we have a flow configuration for which no conductive path can be found from the entrance to the exit. We denote by fn, the set of those nodes nu which have a finite p, assigned by Labyrinth Algorithm 1 (or which have o, = 1 assigned by Labyrinth Algorithm 2), and we denote the set of the

147

20. Maximum Flow

other nodes by 912. In other words, a conductive path exists from the entrance to a node of 9l,, while no such path exists to a node of 912. Then, the set of those internal branches, each of which has one of the endnodesin %, and the other in ?R2, forms a cut separating the entrance from the exit. Let

fi = - b, +

n

4,

b, be the cutset corresponding to the

K=2

segregation consisting of this cut and the reference branch b,. Since the branches of this cut are saturated (i.e., nonconductive) in the respective directions, a branch b, with 4, = 1 carries the flow = cK,and a branch b, with 4, = - 1 carries the flow = b". Therefore, we have

P

P

i.e., the total entrance-to-exit flow g' of this flow configuration { is equal to n

the value

C J;K(q,) of

the cut. Since, obviously, the total flow cannot

K=2

exceed the value of any cut, the g1 attains the possible maximum, and the cut thus obtained has the minimum possible value. REMARK 20.7 An obvious but important fact follows from the solution algorithm. Since the algorithm makes use of numerical computations of addition and subtraction only, the value of the maximum flow as well as that of the flow in each branch obtained by the algorithm must be the result of repeated applications of additions and subtractions to the branch capacities cK and b". (In the general case, the 5"'s must also be taken into 0

account. However, as will be seen in 920.3.2,the 5,'s also can be determined 0

by additions and subtractions of the cK'sand bK's.)Therefore, if the capacities are all integers or rational numbers, then there is a solution (so long as at least one solution exists) consisting of integers or rational numbers, respectively. This fact can also be related with the total unimodularness of a primitive cutset matrix D," (cf. 99.6). The maximum flow problem is formulated in terms of linear programming as the problem of maximizing 5' under the constraints n

cK 2

5" 2 b"

(K = 2,

..., n),

1D""5" = 0

(u = 1 ,

... , m).

K=,

If these constraint relations are rewritten in the standard form, the coefficient matrix, which is essentially the D," augmented by unit matrices of appropriate orders, is also totally unimodular. Therefore, any basis part

148

V. Linear Network-Flow Problems

of the coefficient matrix has the inverse consisting of integers so that every basic solution is obtained by linearly combining the C"'S and bK7swith integer coefficients. 20.3.2. ALGORITHM FOR

THE GENERAL CASE

In this section, we shall consider solving a maximum-flow problem with general c"'s and b " ' ~ Careful . examination of the algorithm in $20.3.1 will prove that it remains valid without any further modification if we take as 6 a feasible flow configuration. (6 = 0 is merely one of the feasible flow 0

0

configurations in case C" 2 0 and 6" i0.) Furthermore, if we regard the maximum-flow problem as a special case of the general linear network-flow problem with branch characteristics such as are indicated in Fig. 20.1, substituting a voltage source for the reference branch, then the algorithm to be developed in $24.1 is applicable. However, a special case can be treated by a special algorithm. We shall give a method for obtaining a feasible 6 which makes use of the algorithm for the special case only once. 0

Our problem may be described to be that of finding a feasible flow configuration in the network obtained from a given two-terminal network with branches having the characteristics of the kind shown in Fig. 20.1 by identifying the entrance and the exit nodes, i.e., by short-circuiting the reference branch. For our purpose, the equivalence transformations of type 2.1 expounded in $19 are useful. We leave untouched those branches for which C" 2 0 and b" 5 0. A branch with b" 2 0 is represented as in Fig. 20.l(b) and then transformed as in Fig. 20.9(a), and a branch with C" 5 0 is represented as in Fig. 20.l(c) and then transformed as in Fig.

q*cq-&y-

*-b ( L 0)

(4

A

(b)

A

-C(ZO)

Fig. 20.9. Modification of a branch.

149

20. Maximum Flow

20.9(b). The middle points A of the split current sources are then combined into one node. We insert in series of each current source thus connected to node A a current limiter of the same value as the source to obtain a network such as that shown in Fig. 20.10(a). The current sources connected to node A are classified into two groups, i.e., those which are directed to A and those which are directed from A , where the sum of the values of the current sources of one group is equal to the sum of those of

7 (b)

Fig. 20.10. Modification of a network.

150

V. Linear Network-Flow Problems

the other. We then remove all the current sources connected to node A by short-circuiting and split A into two nodes, each of which is incident to the current limiters (in series with the sources) of one group. Regarding the two nodes as the entrance and the exit, respectively, we have a two-terminal network such as that shown in Fig. 20.10(b), on which we consider a maximum-flow problem. It is obvious that the maximum flow cannot exceed the sum of the values of the current limiters and that if the maximum flow is equal to that sum, then each current limiter carries the flow equal to its value so that we can easily construct a feasible flow configuration on the network of Fig. 20.10(a) and hence, one on the original network. The converse is also obvious, i.e., if the original network admits a feasible flow configuration, so does the network of Fig. 2O.lO(a), and, consequently, the maximum flow through the network of Fig. 20.10(b) is equal to the sum of the values of the current limiters connected to the entrance (or the exit, which is the same). Since the internal branches of the two-terminal network of Fig. 20.10(b) have nonnegative capacities, we have reduced the problem of finding a feasible configuration in the general case to that of obtaining a maximum flow in the special case. 20.4. Triangular Inequalities A network JV consisting of branches with branch characteristics of the kind shown in Fig. 20.1 may be regarded as different two-terminal networks by specifying in different ways the entrance and the exit nodes, i.e., by connecting the reference branch between different node pairs. Let us denote by cab the value of the maximum flow from n, to na of N , i.e., through the two-terminal network derived from N by specifying node nu as the entrance and nB as the exit, calling it the terminal-pair capacity from nu to nB. We usually define

caa= co

for all a

(20.26)

for convenience.

Theorem 20.2. For any sequence of r nodes n a l , have c,rar

- min(cum > -

, .. . , Ca'-',').

caz~3 3

..., nap (r 2 3), we (20.27)

PROOF. Consider the two-terminal network with n,, as the entrance and nepas the exit. Because of Theorem 20.1, there is a cut separating nar from nap whose value is cUrar.Since the cut separates n, from n, it

151

20. Maximum Flow

separates nai from nai+I for a certain i (1 _I i _I r - 1). Therefore, c ~ ' " ' + ~ cannot be greater than carar,the value of the cut. Hence we have

> - min(c@~a2 -pimi+ I > ,...,

p ~ a r

ca2b3

Y

Q.E.D.

Setting r = 3 in (20.27), we get

- min(cam, >

(20.28)

which sometimes might well be called the triangular inequality for terminalpair capacities. (20.27) for general r can readily be deduced from (20.28). The inequality (20.27) shows that we cannot in general construct a communication network whose terminal-pair capacities are arbitrarily given. In fact, it is known that (see Gomory and Hu [I], Ford and Fulkerson [l], and Berge and Ghouila-Houri [l]) in a symmetric network (i.e., a network in which cK = -b" for every K) with m 1 nodes, at most m out of all the m(m + 1)/2 terminal-pair capacities can have different values. This " triangular inequality " is concerned with the case where the reference branches form a closed path. For the case where the reference branches form a segregation, we have other kinds of inequalities.

+

Theorem 20.3. Let us consider a network JV in which branches b,,, . . ., bKPform a segregation such that r

- b,, +

c

i=2

b,,

(20.29)

is a cutset and of which the branches other than b,, , . . ., bKPare provided with the characteristics of the kind shown in Fig. 20.1 (see Fig. 20.11). Let .Mi be the two-terminal network derived from JV by short-circuiting bK1, .. ., b,,-, , bKi+ .. ., bKr, with b,, as the reference branch, and let

Fig. 20.11. Reference branches forming a segregation.

152

V. Linear Network-Flow Problems

p ibe the value of the maximum flow through N i (see Fig. 20.12). Then we have

c

f"'5

(20.30)

fKi.

i=2

PROOF. Let

- bK; +

.cd i K

(20.31)

bK

K

be the elementary cutset corresponding to the segregation consisting of

(a)

(b)

Fig. 20.12. Illustration of a triangular inequality.

bKi and the cut of the minimum value in the two-terminal network Xi (cf. 520.2). Then we have

f

Ki

=

c 3i"(di,).

(20.32)

K

The sum of the cutset of (20.29) and those of (20.31) with i = 2, (regarded as cutsets in N ) , i.e.,

.. . r ,

(20.33) is also a cutset, in .Af (which may be regarded also as a cutset in .MI), the segregation corresponding to which contains b, in the negative direction, so that the internal branches of the segregation form a cut in N,which is not necessarily of minimum value. [Note here that the diK's in (20.31) vanish for K = K ~ .,. . , K , .] Hence we have (20.34)

2Q. Maximum Flow

153

Since p(q)is convex and linear for the q's of the same sign, we have, for arbitrary q and q',

P(?+ r'>= PC4(2r + 2491 5 %CV(21)+ V(2q')I = Pol)+ V(r').

(20.35)

From (20.34) and (20.35), it follows that

The special case (r = 3) of (20.30), i.e., f"'

f"' + f K 3 ,

is of the same form as the ordinary triangular inequality. It will be made clear in 521.4 that it is in fact the "dual" of the ordinary triangular inequality for distances. 20.5. Applications

Besides the obvious applications, such as the problem of planning the transportation of as much commodity as possible through a network of roads with limited transportation capacities or of sending as much information as possible through a network of data-communication channels with limited information transmission rates, there are a great variety of applications of the maximum-flow problem. This section is devoted to the review of those applications. RELATIONS 20.5.1. SUPPLY-DEMAND

This problem concerns a kind of multiterminal network. The network JV with which we are concerned consists of branches provided with the characteristics of the kind somewhat more special than that in Fig. 20.1, i.e., in which the amount of flow is restricted by 05

ly

5 CK,

(20.36)

with given cK's.(It is not very difficult, however, to extend the following arguments to the case where the lower bound for 5" is 6" instead of 0.) Among the set of nodes % of JV we specify two mutually disjoint subsets asand S t ,calling their elements entrances and exits, respectively. Through a node na of %, a flow xa goes from the outside of N into N , through a

154

V. Linear Network-Flow Problems

node n, of %, a flow y" goes to the outside of N , and at every node of % - (YlS u %J no flow goes in or out between N and its outside. We further require xa's and ya's to satisfy 0 5 sa 5 xu 5 sla

(n,

E 'ills),

5fa

(n,

E %J,

0 5 t a 5 y"

(20.37)

where the sa, s'", t', and t" terms are given quantities. Our problem, called a supply-demand problem, is to find a set of conditions under which there exists a flow configuration in JV satisfying (20.36) and (20.37). This problem has an obvious application where we regard N as a transportation as the network with routes of limited capacities (20.36), a node of location of a producer of a commodity, a node of %, as the location of a consumer, and [s", s'"] and [t", t'"] as the desired ranges of production and consumption of the respective producers and consumers. The above problem can easily be reduced to the standard form we have so far investigated as follows (cf. Fig. 20.13). We prepare new branches,

(a)

(h)

Fig. 20.13. Supplydemand network.

one corresponding to each of the nodes of 8, u %,, and a new node n o , in addition to those already existing i n N , and connect the branch correbetween the nu and the new node no sponding to a node n, of XS(or a,) in the direction from no to na (or nmto no), providing the new branches with the characteristics of (20.37) where the x"'s (or y " ' ~ are ) regarded as the flows in the new branches. We shall denote the network thus obtained by calling the new branches and the new node the additional branches and the ndditiotml node, whereas we shall call those of JV the intrinsic braticlies and the intrinsic nodes. The supply-demand problem of N is

J,

155

20. Maximum Flow

evidently equivalent to the existence problem of a feasible flow configuration of j. Here we note a relation (20.38)

y.

which holds for any flow configuration on Now we shall investigate what results are obtained if we apply Theorem 16.5 in $16.4 to $. Let q be an elementary cuset on 2.If car q contains no additional branches, we have always

-

Wq)

e 0,

since, for an intrinsic branch,

V($e 0 because

V(v])= cKil 2 0

if

v]

20

and

J;"(v]) = bKv] = 0 if

v]

5 0.

If car q contains additional branches, we consider the dissection into two subsets of the node set % u {no} of 2 corresponding to q (cf. $9.1), denoting the dissection of % u {no} by u {no}, %,) or (!R1, %, u {no}) according as to which of the two subsets the additional node no belongs. Let us further introduce the notation T(YIl,%,) to denote the sum of the cK's of the intrinsic branches which start from a node of 8,and end at a node of %, and s ( ' W , s'('%), s(a2), s'(%,), W,),Wl),t W 2 ) , f'(%,) to denote the sum of sa, s'", fa, t" of the additional branches corresponding to the nodes of %, n %,, %, n %, %* n Sl, 92,n %, respectively. All these quantities are nonnegative by definition. As is seen from Fig. 20.14, in case q dissects % u {no} into (Sl,8,u {no}), we have

-

Wq) = TWl, a,)- s W 1 )

+ t'(%l).

(20.39)

Similarly, in case the dissection is (91, u {no}, %,), we have

-

wl)= T(%,

%2)

+ s'(W - @2).

(20.40)

Thus we have Theorem 20.4. For the require'ments (20.37) of supplies and demands to be satisfied by the network JV with branches of capacities cK, it is necessary and sufficient that for any dissection (sl,%,) of the set of nodes

156

V. Linear Network-Flow Problems

Fig. 20.14. Supply-demand theorem.

91 of A?, the relations

hold. The necessity of the conditions (20.41) might intuitively be evident, since the first inequality states that .N should have at least the capability of transporting the excess of the lower bound of the total amount s(Yt,) of commodity produced in 'illl over the upper bound of the total amount t'(%,) consumed in %,, and similarly for the second inequality. What interests us is the fact that the conditions are sufficient. It is worth noting that, if we set

sa = 0

for all

n,E

%,s,

t" = 00

for all nu E 'illr, (20.42)

then the first inequality of (20.41) holds identically so that only the second is necessary and sufficient, whereas, if we set stu = co

for all

na E !TIs,

la = O

for all n u € gr, (20.43)

then the second one holds identically so that only the first is necessary and sufficient. Therefore, we see that the original problem may be decomposed into two problems (20.42), (20.43) of more special kinds.

20. Maximum Flow

157

Two examples from among the many problems that can be formulated as supply-demand problems will be shown in the following. The first is the so-called " boys-and-girls problem." Suppose that there are a number of boys and the same number of girls among whom the relations of friendship are defined, i.e., any couple of a boy and a girl are, by definition, either friends or not. The problem is to examine whether or not it is possible to establish a partnership in such a way that each of the boys and girls may participate simultaneously in dancing with one of his or her friends. The network representing this problem may have the following structure (see Fig. 20.15). It has one node corresponding to each boy and one node to each girl, the nodes of the former forming the set asand :ndr

Fig. 20.15. Boys-and-girlsproblem.

those of the latter a,, and it has no node other than = asu %,. It has a branch with capacity c" equal to 1 connecting a node of asto a node of a, if and only if the boy corresponding to the former node and the girl corresponding to the latter node are friends. (More generally, the following arguments will not be affected even if we take as the capacities of branches an arbitrary value greater than or equal to I .) Furthermore, the 8''s and the S'"S of n, E as are all put equal to 1, and so are all the t "s and the r '"s of na E %,. The network thus defined admits an integer solution if it admits a solution at all (cf. Remark 20.7 in §20.3.1), where it is noted that in an integer solution, the amount of flow in a branch is equal to either 0 or 1, since it should lie between 0 and 1 (= cK). (If we put C" > 1, then we have to derive this fact from the condition that the sum of the flows in the branches incident to a node of as[or a,] should be equal to 1 .) We may thus conclude that a branch has a unit flow if the boy and the girl corresponding to its two endnodes are combined to form a couple, and that it has a null

158

V. Linear Network-Flow Problems

flow if they are not. I n this way, the problem is reduced to a supply-demand problem, which can be solved by means of the algorithm in 920.3.2. Since the intrinsic branches in the network representing this problem have nonnegative capacities and the additional branches are mere current sources (s’ = s” = 1, t’ = t’‘ = I), the transformation shown in Fig. 20.10 is highly simplified. I t can readily be shown that if each boy has in girl friends and each girl has 111 boy friends, then the problem has a solution, i.e., all the n boys and the 17 girls can participate simultaneously in dancing. In fact, we have an obvious noninteger solution for the corresponding network problem in which every intrinsic branch carries I/nz unit of flow, so that the problem has an integer solution (cf. Remark 20.7). As the second example, we take up the famous theorem by G. Birkhoff and J. von Neumann:

J?

A doirbly stochastic matrix cat? be expressed as a itveighted mean of pertnutation matrices,

where by a doubly stochastic matrix of order n we mean a matrix P = (pup) (a, P = 1, . . . , n) such that (20.44) and by a permutation matrix we mean a doubly stochastic matrix whose elements are 0 s and 1’s. To prove this theorem, we consider a solution of the boys-and-girls problem where the ath boy and the Pth girl are friends if and only ifp,, > 0 in the given doubly stochastic matrix P. The solution, if it exists, may be expressed by a permutation matrix Q = (quo),where qup= I if the ccth boy and the Pth girl are combined to form a couple and equals zero otherwise. Except for the integrity of the qua elements, the conditions to be satisfied by such a Q are (20.44) and (20.45): 4,s > 0

only if Pup > 0,

(20.45)

and it is obvious that Q = P satisfies these conditions. Therefore, by virtue of Remark 20.7, we have an integer solution Q , i.e., a permutation matrix satisfying (20.45). Let puopobe the smallest among the pupelements of P for which qup > 0. Then, as is easily seen, we can set (20.46) P = ~ n ~ p+ ~(1 Q- P U , ~ ~ ) R , where R is doubly stochastic and has a smaller number of nonnull elements than P has. Continuing the similar decomposition for R,we can obtain the required expression of P.

159

20. Maximum Flow

20.5.2. THEORY OF MATCHING In this section, we shall deal with the combinatorial-mathematical problems connected with the names of Konig, EgervBry, Menger, Hall, et al. The typical problem of matching is formulated as follows. We consider two mutually disjoint sets '3, and '3, and a binary relation p defined between the elements of '3, and those of '3, (Le., p specifies a subset of 9Isx a,), and we need to choose as many pairs ( n a , ns) as possible such that n a p nfi,

"a€'%,

BE'^,,

and such that no element of '3, or '3, is chosen in more than one pair, where na p nB is read as " nu in relation of p to ns." In terms of a matrix, the problem may be stated as follows. Let P = (pus)be a matrix with rows corresponding to the elements of '3, and columns to those of W,, in which pas = 1 if nu p ns and equals zero otherwise. Then we require to choose as many nonnull elements (i.e., 1's) in such a way that from each row (and each column) at most one 1 is chosen. We shall call a set of pairs or 1's satisfying the foregoing requirement of choice an admissible or a matching set. Thus, our problem may be said to be that of obtaining a maximum admissible set. Like the supply-demand problem in the preceding subsection, the matching problem can also be formulated as a maximum-flow problem. The structure of the two-terminal network Jlr is as follows. In addition to the nodes corresponding to the elements of Wsu '3,, N has two nodes n, and n , playing the roles of the entrance and the exit, respectively. JV has a branch connecting n, to each node of '3, (with unit capacity in the positive direction and null capacity in the negative direction), a branch connecting each node of '3, to n , (with unit capacity in the positive direction and null capacity in the negative direction), and a branch connecting a node na of '3, to a node nB of '3*if na p nB (with infinite capacity in the positive direction and null capacity in the negative direction); and it has no other branches. We shall call the branches of the first and the second kind additional branches and those of the third kind intrinsic branches (see Fig. 20.16). The integer solution of the maximum-flow problem for JV corresponds to the solution of the matching problem in an obvious way where the intrinsic branch corresponding to a chosen pair ( n a , ns), or the chosen pap = 1 , carries the unit flow, and the total flow from n, to n , gives the maximum number of pairs of an admissible set.

160

V. Linear Network-Flow Problems

Fig. 20.16. Matching problem.

Let us now investigate what the maximum-flow minimum-cut theorem (Theorem 20.1, 420.2) will give in the present case. Since the value of the maximum flow through the two-terminal network JV now under consideration is evidently finite (in fact, it cannot exceed the cardinality of Ws or %,), we have to search for a minimum cut among the cuts of finite values. If a cut separating ns from nr dissects the node set % = asu 'illru {ns, n,} into W, and 8,(n, E %,, n, E W2) and if there is an intrinsic branch contained in the cut in the positive direction, i.e., one which starts from a node of W, n W, and ends at a node of W tn W2, then its value is infinite. Therefore, a minimum cut should contain no intrinsic branch in the positive direction, but it should consist of: (1) branches connecting ns to the nodes of 'S2n (contained in the positive direction), (2) branches connecting the nodes of W l n %, to n, (contained in the positive direction), and (3) branches connecting the nodes of %, n 9Is to those of W l n W t (contained in the negative direction),

where its value is equal to the sum of the numbers of the branches of the first and second groups (cf. Fig. 20.16). In order to express a cut of this kind, it is convenient to introduce the concept of covering set. A covering set Wc is by definition a subset of Wsu %, such that every intrinsic branch has an endnode in it, or, in terms of the original form of the matching problem, that if na p np , then either

161

20. Maximum Flow

n, E Wcor nB E Wc. In terms of the matrix representation, a covering set is a subset of rows and columns such that every 1 is on one of those rows and columns. We can establish the one-to-one correspondence between the covering sets and the cuts in N of finite values as follows. For a given cut associated with the dissection (al,W,), we set

W, = (92, n W,) u (Wln at).

(20.47)

Since the intrinsic branches connect the nodes of W, = (Wln 8,)u (W, n W,) to those of W t= (Wln W J u (Wzn 8J,whereas there is no intrinsic branch connecting Wl n W, to 8, n Wr,every intrinsic branch should have an endnode in (%, u %,) u (TIl n !TIt). Conversely, for a given covering set Sc, we set

8,= {ns} u (Wcn Wr)u (W,- W, n WJ, %, = In,> u

(’% n 8,) u (W,- 8,n W,

(20.48)

defining the cut associated with the dissection (Wl, WJ.Then we have

(8, n 8,) n Wc= (W, n %J n W, = 0.

(20.49)

There is no intrinsic branch connecting W ln W, to W 2 n !Rr, for, if one such branch existed, one of its endnodes would be in W,, in contradiction to (20.49). In the foregoing terminology, we can restate Theorem 20.1 for the present problem as follows. Theorem 20.5. In a matching problem, the number of pairs of a maximum admissible set is equal to the number of elements of a minimum covering set, where by a ‘‘maximum (or minimum) set” we mean a “set with the maximum (or minimum) number of elements.’’ A practical means to obtain a maximum set of pairs as well as a minimum covering set is afforded by applying the algorithms in $20.3.1 to a network of the kind illustrated in Fig. 20.16. The computations are naturally simplified due to the special network structure of the problem. It will be of use to expound the simplified form of the algorithm in terms of the matrix representation. (The reader should make detailed comparison of the following algorithm with the algorithm in $20.3.1. In the following algorithm, the intrinsic branches are treated as if they had unit capacity instead of infinite capacity, which, as will easily be seen, has no influence on the problem other than simplifying the algorithm.)

162

V. Linear Network-Flow Problems

Algorithm for a Matching Problem. To each row (corresponding to an element of a,) as well as t o each column (corresponding to an element of ’ill,), we assign a signed number +pa, and we encircle some 1’s of the matrix (which correspond to the intrinsic branches carrying unit flow). We start from the given matrix with none of its elements encircled, and then increase the number of encircled l’s, one by one, according to the following rules (i)-(iii): (i) Set f pa = -0 for every row nu having no encircled I , fPa = - 00 for every column nu having no encircled 1, and +Pa = co for all the other rows and columns. (ii) Repeat (iia) and (iib) alternately so long as it is possible. (iia) We take up a row nu whose pa has a minus sign, convert the minus sign before the pa into a plus sign; and, if there is an unencircled 1 in the row nn and if that 1 is in a column nu, with +pa. = -co, then we go on to (iii); if there is an unencircled 1 in the row nu and if that 1 is in a column nu, with +pa. = +co, then we convert the fp,. = +co into +pa, = - a , and we may try another row or go on to (iib). (iib) We take up a column nu whose pa (Zoo) has a minus sign, convert the minus sign before the /?, into a plus sign; and, if there is an encircled 1 in the column nu and if that 1 is in a row nap with +pa. = + co, then we convert the +pa. = +co into +pa. = -a. Then we may try another column or go on to (iia). (iii) Repeating the operations of (ii), we shall encounter the situation in which we either find a column with no encircled 1 [as it happens in (iia)], or we cannot decrease the number of the rows and columns with i-pa = + co any further. I n the latter case, the encircled 1’s form a maximum admissible set of pairs, and the rows with +pa = + co together with the columns with i-& # +co form a minimum covering set. In the former case, let the column finally attained be nao.Then we determine the sequence of rows nu,, n u , , . . . and columns nar, na4, . . . by

+

@i+l

=Psi

(20.50)

to arrive at Pa,,,,

=o

for a certain r. Due to rule (ii), the ( u 2 i + l ,uzi)elements of the matrix are unencircled 1’s ( i = 0, . . . , r ) , whereas the ( u 2 i - l , uZi) elements are encircled 1’s (i = 1, . . . , r). We encircle the former 1’s and remove the circles around the latter 1’s.

163

20. Maximum Flow

EXAMPLE.To find a maximum admissible set of l’s, as well as a rflinimum covering set of rows and columns, of the matrix (20.51), where we number the rows 1-3 and the columns 4-6:

Fl. 4 5 6

: 3

tc

*pa

1 -0

1 0 0

2 -0

4

3 -0

--a3

5

6

-a! - m .

(20.52)

(& = 0, r = 0).

(20.53)

-1

+O

a,, = 4,

(20.51)

a1 = p4 = 1

Thus the 1 to be encircled is the (1,4) element alone. The resulting matrix is shown in (20.54):

(20.54) 3

1 0 0

For the matrix of (20.54), we perform operations similar to the above, i.e., we determine the initial values of the f P a ’ s as shown in the second line of (20.55), modify them as shown in the third line, and then stop, to obtain

164

V. Linear Network-Flow Problems

the ai’s shown in (20.56) as well as the matrix of (20.57): a

+&

1

2

3

4

5

foo

-0

-0

+co

-co

+O

6

(20.55)

-00,

-2

4 5 6 @ I 0

(20.57)

0 0 1 1 0 0

For matrix (20.57), the initial values of the +pa’s are determined as in the second line of (20.58). Finding +p3 = -0 having a minus sign and an unencircled 1 in row 3 in column 4, we change +p3 and +p4 to + O and -3, respectively, as shown in the third line of (20.58). Then we find an encircled 1 in column 4 with +p4 = - 3 in row 1 to modify +p4 and +PI as shown in the fourth line. Repeating similar processes, we finally arrive at column 6 with + / j 6 = -- 00. The ai’s are determined as shown in (20.59), starting from a. = 6: a 1

1

2

3

4

5

6

+00

+co

-00

-3

I

-4

I

+4

+3

Y

(20.58)

-1

I

-5

I

+5

-2

Thus we add a circle to the (2, 6), (1, 5), and (3,4) elements of the matrix

20. Maximum Flow

165

of (20.57) and remove the circle from the (2, 5) and (1,4) elements to have (20.60): 4

5

6

I

I

1

1 0 0

2

0

1

0

.

(20.60)

The initial values of the &pa’s for the matrix (20.60) are as shown in (20.61), for which we can no longer perform any modification operations : 1

(20.61) Therefore, we have obtained a maximum admissible set of 1’s (those encircled) in (20.60). Gathering the rows with pa = GO and the columns with pa # GO, we obtain a minimum covering set, i.e., the set consisting of rows 1 , 2, and 3. Also famous is the following theorem in the theory of matching, where we make use of the notation 11?311 to denote the cardinality, i.e., the the subset of %, number of elements, of the set 9 , and we mean by r’(%’) consisting of those elements each of which is in relation of p with some element of a given subset %’ of SS,i.e.,

r“(W)

= {nB I nBE ‘91t,

3 na E W( s % ~na)p: np}.

(20.62)

Theorem 20.6. A matching problem has a solution in which all the elements of ‘91sappear in the chosen admissible set if and only if for every we have subset %’ of

II%’II 5 lP-+(WIl.

(20.63)

PROOF. The necessity of condition (20.63) is too obvious to require any explanation. For sufficiency, we have only to show that if some elements of ‘91sare not chosen, then there is a subset %‘ of 9lssuch that

Il%’II > llr+(~’)ll.

(20.64)

In fact, Theorem 20.5 assures the existence of a covering set ‘iRCsuch that

II%II < II%sll

(20.65)

166

V. Linear Network-Flow Problems

in such a case. If we set %‘ = 9Is - %, n

(20.66)

r+(%’) 5 aCn a,,

(20.67)

we have due to the definition of the covering set. From (20.66) follows the relation

Il%’II

=

II%ll - 11%

n WI,

(20.68)

and from (20.67),

IIr+(%’)II 5 II%c

(20.69)

n

Subtracting (20.69) from (20.68) and taking account of the identity %c

= (%c n

as) u

n

we have

IIa’II - IIr+(fi’)II L IIPII - ( I I m c n WI + IIytc n %II> = Il%sll - I I ~ U *

(20.70)

(20.70), combined with (20.65), yields the required inequality (20.64). Q.E.D. Theorem 20.5 can be extended in two ways, as follows, in regard to the underlying network structure. Let us consider a graph with arbitrary topological structure in which two mutually disjoint subsets, %, and of the set of nodes % are specified and each of whose branches is defined to be conductive in the positive or negative or both directions. A path from a node of asto one of S t is called, as before, a conductive path if every branch is conductive in the direction in which it is contained in the path. A set of such conductive paths is said to be branch-admissible if they have no branch in common, and node-admissible if they have no node in common. Then, the problem is to find a maximum branch- or node-admissible set of paths. The basic form of the matching problem, as we have treated so far, may be considered a special case of the problem of obtaining a maximum branch-admissible set (where the underlying graph is the entire graph of the kind shown in Fig. 20.16 consisting of intrinsic branches and of additional branches, and the subsets Yls and in the latter problem are taken as the single nodes ns and n, , respectively, in the former), or as a special case of the problem of obtaining a maximum node-admissible set (in which the underlying graph is that part of the graph of the kind shown in Fig. 20.16 which consists of intrinsic branches alone, and the ‘8, and in the former problem themselves are the and Srin the latter).

20. Maximum Flow

167

The problem of obtaining a maximum branch-admissible set as well as that of obtaining a maximum node-admissible set can be translated into a maximum-flow problem by means of the same technique used previously when we translated the supply-demand or matching problem into a maximum-flow problem. The prohibition of a common branch is accomplished by imposing the capacity restriction ojg"s1,

(20.71.1)

-ls("so,

(20.71.2)

-1Sg"Sl

(20.7 1.3)

or

on the flow g" in a branch b, which is conductive in the positive or negative or both directions, whereas the prohibition of a common node (note that this automatically implies the prohibition of a common branch) is accomplished by giving to each node the unit capacity (which in turn is realized by a suitable modification of the network structure such as illustrated in Fig. 20.3). The maximum-flow minimum-cut theorem, Theorem 20.1, when applied to such a network, yields the following theorems, the details of the proofs of which are left to the reader. Theorem 20.7. The number of paths of a maximum branch-admissible set is equal to the number of branches of a minimum branch-cut, where a branch-cut means such a set of branches that, if we open-circuit all of them, we can no longer find any conductive path from a node of !TIs to a node of !TIt. Theorem 20.8. The number of paths of a maximum node-admissible set is equal to the number of nodes of a minimum node-cut, where a node-cut means such a set of nodes that, if we open-circuit all the branches incident to them, we can find no conductive path from a node of !TIsto a node of !TIt.

The theory of matching has a large number of applications to combinatorial mathematics and we shall take up two, i.e., the systems of distinct representatives and the decomposition of a partially ordered set. The problem of the system of distinct representatives is stated as follows. We consider a finite set { E ~ E, ~ .., .} and a finite number of its subset {ol, 02,. . .}. An element of oimay be specified as the representative of the

168

V. Linear Network-Flow Problems

a,. The problem is to choose a representative for each subset such that the representatives thus chosen are mutually distinct. This problem can be reduced to a matching problem by putting !TIs= {a,},!TIt = {ci} and writing a,p c j if and only if e j belongs to ci,as will be obvious. For example, Theorem 20.6 may be restated as follows:

A necessary and sufficient condition for the existence of a system of distinct representatives is that the union of an arbitrary number of subsets aicontains at least as many distinct elements as that number.

The proposition given at the end of the descriptions of the boys-and-girls problem in $20.4.1 may be rephrased as follows: A set of distinct representatives exists if each of the subsets ai contains exactly m (>O) elements and each of the elements c i belongs to exactly m subsets.

The problem of distinct representatives has many variations which are easily translatable into network-flow problems. For the details of such variations, we suggest that the reader consult Ford and Fulkerson [l]. The second problem we shall deal with concerns the decomposition of a partially ordered set that was first treated by Dilworth. A partially ordered set 9l = {n,} is by definition a set provided with a relation (called a partial order) and satisfying

+

n,

+ n,

for every n, E 9l,

> n,,

if

n,+

n,

and

n,

if

n,+

n,

and

n,>

n,,,

(20.72.1)

then

n, = n,,

(20.72.2)

then

n,$

ny.

(20.72.3)

It will be noted that in the problem of project scheduling in 42 of Chapter I, the jobs form a partially ordered set with regard to their precedence relation. We construct a graph by regarding the elements of a given partially ordered set as the nodes of the graph and assuming the graph to have a branch from n, to n, if and only if n, > np (i.e., n, $ n, but n, # n,). The graph thus constructed is characterized by the property that it has no closed path in which every branch is contained in the positive direction. Conversely, we can construct a partially ordered set from a given graph having such a property. Usually we do not write explicitly those branches representing the order relation that follows from others by means of (20.72.3), i.e., we omit a branch from na to nB, if na $ n, and if there are

169

20. Maximum Flow

nu,, ..., n., such that the n a , nB, and nui'sare mutually distinct and such n., 3 nB. However, in the following exposition we that nor3 nu, 3 * shall take account also of such branches for the sake of convenience. (The results obviously remain valid, however, if those branches are omitted.) We call a sequence of r distinct elements (nu,, .. . , nu$ of a partially nu, a descending chain of length r. In ordered set such that nor, particular, we regard a single element as a descending chain of length 0. Here we pose the problem: Find the minimum number of mutually disjoint descending chains such that every element of % belongs to one of them. In the graphical terminology, a descending chain of length r is an elementary path of length r - 1 such that all its branches are contained in the positive direction, and the problem is to find the minimum number of such mutually disjoint paths covering all the nodes. In terms of the precedence relations among jobs in a project, the problem is to find the minimum number of workers necessary for completing the project, where we assume that a worker can do a sequence of jobs that forms a descending chain in the above sense. Let us use the graphical terminology in the further consideration. A descending chain of length r corresponds to a path of length r - 1 which contains r - 1 branches. Therefore, if a partially ordered set with n elements is decomposed into rn mutually disjoint chains, then the corresponding paths contain n - m branches in all. Hence, we may search for a set of paths (with the desired property) whose total number of branches is as large as possible. The subset of branches contained in a set of paths (with no common node) can be characterized as a subset wherein each node of the graph is positively incident to at most one of the branches of the subset and negatively incident to at most one of the branches of the subset. Thus, the problem is reduced to that of obtaining a subset of the kind having as many branches as possible. This problem is equivalent to the following matching problem, as is easily seen. We consider two replicas %+ and %- of the set of elements % of the given partially ordered set, denoting by nu+ or nu-, respectively, the elements of %+ or %- corresponding to the element na:of %. Between the elements of %+ and those of %-, we define the relation p such that

-+

+ + -

nu+ P "6-

0

.

if and only if nu 3 nB and

nu # nB.

Then, the problem now under consideration may be regarded as the and p ; i.e., we must obtain a maximum matching problem for %+, 8-, admissible set of pairs (naf, ns-). In fact, as is easily seen, an admissible

170

V. Linear Network-Flow Problems

set of pairs of %+ x %- corresponds to a set of mutually disjoint descending chains in %, and vice versa, if we make a pair ( n u + , n s - ) correspond to the branch connecting nu to ns . This reformulation of the problem allows us to apply Theorem 20.5. Let the maximum number of pairs or branches chosen in the foregoing be n - m o, i.e., let the minimum number of mutually disjoint chains be m, . Theorem 20.5 states that the number n - m, is equal to the number of elements of a minimum covering set which is a subset of %+ u %-. Obviously, Ilfncll 5 min(Il%+II, Il%-Il)

=

11%+II

=

11%-II

=

11%211

= n.

We denote by SUthe subset of % consisting of such elements of % that neither their correspondents in %+ nor their correspondents in %- belong to Sc. By virtue of the definition of a covering set, there is no pair ( n u + , n p - ) such that nu+ p n p - ,

nu+ E %+

-

nD- E %- -

aC.

Therefore, no two distinct elements of flu are comparable, i.e., if n u , np E flu and nu # n p , then we have neither n u + np

nor

np

>nu.

Let us consider a maximum subset '9Iv (of %) whose elements are mutually noncomparable. Obviously, we have

Il%ull 5 Il%ll 5 mo.

(20.73)

Next, in case n = 11%11 > 1 (the trivial case n = 1 may be excluded), let us consider the correspondence between % and %., For each nu E 91 - %", either nu+ or nu- belong to '3, due to the definition of g U . But it is impossible for both nu+ and nu- to belong to gC.[In fact, if nu+, nu- E %,-, then there is an element n y - E %- distinct from nu- such that nu+ p n y - ;for otherwise, all the elements of %- which are in relation of p to nu+ would belong to a,, which would mean that - { n u + } could be a covering set in contradiction to the assumption of the minimality of '3,. Similarly, if nu+, n,- E fl,, then there is an element n8+ E %+ - SC distinct from nu+ such that n d + p n u - . The elements n u , n y , nd of % corresponding to these nu', n y - , n d + then satisfy the relations nu+ny,

which imply nd

%>nu,

> n y or n d + p n y -

.

(20.74)

20. Maximum Flow

171

(20.74) contradicts the assumption that gCis a covering set.] Therefore, we have

11%

- %,I1

=

Il%cll = n - m,

or

Il%,l

=n

- (n - m,) = m,.

(20.75)

Combining (20.73) and (20.75), we have:

Theorem 20.9. The minimum number of mutually disjoint descending chains containing all the elements in a partially ordered set is equal to the number of elements of a maximum subset whose elements are mutually noncomparable. 20.5.3. APPLICATIONS TO GRAPH THEORY It sometimes happens that a purely graph-theoretic problem can better be treated by transformation into a network-flow problem. Theorems 20.7 and 20.8 are two such problems. We shall give another example. The problem is to find a subgraph sj of a given graph 6 (i.e., a graph sj obtained from 6 by open-circuiting some branches of 6)such that to each node na of 6, the number of the positively incident branches of sj lies between sa and s'" and the number of the negatively incident branches of fi lies between t" and f a ,where sa, s'", t", and t'" are given numbers. This problem can be reformulated into a supply-demand problem as follows. We shall make use of a technique analogous to that which we have resorted to in order to formulate the problem of decomposing a partially ordered set into descending chains. We consider a network Jf having the set of nodes %+ u %- where %+ and %- are two replicas of the set of nodes % of the given graph 6. We assume that Jf has the same set of branches 23 as 6 has (exactly speaking, the two sets are put in one-to-one correspondence) and that, if a+b, = n a , 8-b, = nB in 6, then a+b, = na+, 8-b, = nB- in N , where na+ is the correspondent in %+ of n, in % and nB- is the correspondent in %- of nB in %. Furthermore, each branch of N is provided with the unit capacity in the positive direction and the null capacity in the negative direction. If the %+ and %- are regarded as the set of entrances and exits and if sa, s'" and t", t" are regarded as the lower and the upper limits of the supplies at the entrances and the demands at the exits, then we have a supply-demand problem (cf. 820.5.1). The relation between this supply-demand problem and the subgraph problem now in question is as follows. If a subgraph fi of 6 satisfying the required conditions is known, then we can obtain a solution of the supply-demand

172

V. Linear Network-Flow Problems

problem by assigning the unit flow to each of the branches of N which correspond to the branches of fi. Conversely, if an integer solution of the supply-demand problem is known, then we can construct fi by opencircuiting those branches of 0, which correspond to the branches of N carrying null flow. In this way, the subgraph problem now in question can be reduced to a supply-demand problem. Let us consider in more detail the implications of Theorem 20.4 in this case. In regard to a dissection (!Ill, 9Iz) of %+ u %-, we shall make use of the abbreviations

= g1n %+, flz+ = %, n %+,

G %l

n %-,

flz-= ’illz n %-.

Due to the special structure of N , the quantities appearing in (20.41) may be interpreted in the present case as follows: -

“(fl,, f12)

= the

of

number of branches connecting the nodes + to those of

s(%,) = s(%, +),

s’(%z) = S ’ ( S Z + ) ,

t(fll) = t(fll-),

t’(%z) = t ’ @ - ) .

Hence, if we denote by aZ-)llthe number of branches in N conthen (20.41) becomes in the necting the nodes of 9Il+ to those of SZ-, present case :

Il(%l+, 11(%+,

2 e l + ) - t’(%-), h w 2 - ) - s‘(%+).

fl2-111

%z-)ll

Let us then consider the minimum value of . We have fixed

!R2-)ll

(20.76)

+ t’(%,-) with

+

ll(%+, ‘%)I1

+ t’(fl1-1

= nB-

2

1 ll(%l+, “/J->ll +nB- c t’B 1 min[II(%l+, ns-)ll, t B ] (20.77) ~911-

~g12-

“ 8 - E%-

np-)ll means the number of branches b, such that a+b, E 9Il+ where for a given fll+ such that and 8-b, = n p - . If we choose s 1 -

= {np-

I f P < Il(%+,

n/J-)ll>,

(20.78)

then (20.77) holds with the equality instead of the inequality sign. Therefore, the requirement that the first inequality of (20.76) should hold for an

173

20. Maximum Flow

arbitrary dissection (illl,a2) of '3' u 3-is equivalent to the requirement that 1 minCII(Wl +,np-)II, t"1 2 s(9tl + I , (20.79) na-sm-

should hold for an arbitrary subset 'illl+ of %+. (20.79) may be rewritten in terms of quantities in Q as (20.80) np)ll means the number of where Sois an arbitrary subset of % and branches b, of 6 such that d + b, E 'illo and d - b, = nD . Similarly, the requirement that the second inequality in (20.76) should hold for an arbitrary dissection ('illl, 9L2)of %+ u W - is seen to be equivalent to the requirement that C minCll(n,, %o)ll, 2 tp (20.81)

c

AT'"]

n p E%

n,E%

should hold for an arbitrary subset Soof % (in (t), where / [ ( n u ,%,)I] means the number of branches b, of 6 such that d + b, = nu and 8- b, E %o. Thus, we have seen that the validity of (20.80) and (20.81) for an arbitrary subset Wo ( & %) is necessary and sufficient for the existence of a subgraph Sj with the required property. [The reader should prove the necessity of (20.80) and (20.81) in a more direct way.] In case s" = s'"

and

for all a,

t" = t'"

(20.82)

the preceding conditions can be somewhat simplified. In this case, it is obviously necessary that

C s" = rips% C tp,

i.e.,

s(%+)

= t(%-).

(20.83)

n,sB

Since (20.82) and (20.83) imply $('ill1

+) - t ' ( % l - ) = s(%1 +) - t(%n, -) = {S(W+) = t(%,

-)

- s(W,')}

- { t ( K ) - I(%,-))

- s(9I2 +) = t(!R2-) - S'(%,J,

it follows that the validity of (20.83) together with that of either (20.80) or (20.81) for an arbitrary W, is necessary and sufficient for the existence of a subgraph Sj. A factor of a graph is usually defined as a subgraph which is the union of elementary closed paths which have no node in common and in which

174

V. Linear Network-Flow Problems

the branches are contained in the positive direction. It will be seen without difficulty that, in our terminology, a factor may be defined as a subgraph J3 with sa = S" = t a = t'" = 0 or 1 for every node n a . If a factor @ which contains all the nodes of Cfi is required, we may put sa = d a = t a = t'" = 1 for all a. In that case, it readily follows from the above considerations that: a necessary and sufficient condition for the existence of a factor containing all the nodes is that for an arbitrary '$lo ( E %), we have

llr+(%o)ll 2 II%olI

3

(20.84)

where

r+(%J = (J r + n , . n, t S o

[The reader should derive (20.84) from (20.80) or (20.81).] It should be noted that the practical algorithm for obtaining a subgraph or a factor with the required property is naturally derived from that for the maximumflow problem when we apply the latter to the respective networks. Sometimes we encounter the problem of finding a factor containing as many nodes as possible. In this case, we may modify the network N constructed above as follows.'For each a, we add a branch which connects nor- to na+ and which has the unit capacity in the positive direction and the null capacity in the negative direction, to represent the constraint sa = - t a = t" = 0 or 1. On the network thus modified, we may search for a flow configuration such that the sum of the flows in the newly added branches is as large as possible. In fact, the sum represents the number of nodes contained in the factor corresponding to the flow configuration. If we assign the convex function q,(t) = 0 to each branch of .,If and q,(x)= --x to each of the newly added branches, then the problem is to find a flow configuration that minimizes the sum @ of those functions. The problem of obtaining such a flow configuration is beyond the scope of this section, but it will be dealt with in $24. (Note that here we are assuming the validity of Remark 20.7 under somewhat more general circumstances.) 21. MINIMUM ROUTE

21.1. Presentation of the Problem The problem to be dealt with in this section is formally dual to that in $20. It is to obtain the projection to the q axis of the two-terminal characteristic curve of a two-terminal network .,If (6;ref., C,, . .. , C,) for which each C, (K = 2, . .. ,n) is a characteristic curve such as shown in

175

21. Minimum Route

Fig. 21.1. In a more natural expression, the problem is to obtain both the maximum and the minimum values for qI (the tension across the reference branch, i.e., the difference of the potentials at the entrance node and at the exit node) when the tensions qKacross the internal branches (K = 2, .. ., n) are subject to the constraints:

v, E

or dK6 qK 5 eK

JK

(J, = Cd,, e K l , e, 2 dK).

In the foregoing, it is understood that we put eK= co and/or d, when there is no upper and/or lower bound for qK. b,

= - co

-

b, e, - (1,

~

(21. I )

+fyJ+ c4tf'> =

- d,

- (IK

(4 b,

=

(b)

t-

+ e k q4 q + e, -

(d b, =

*

#<>- - ?

$*7?-+- (1, (d )

(e)

Fig. 21.1. Spanned branch.

We can specify the problem also in terms of nodes and node potentials as that of obtaining the maximum (as well as the minimum) possible value of the difference 5, - 5, between the potentials at the entrance ns and the exit n t when the node potentials [, satisfy

d, 5 la-

c8 5 eK

if ab, = nor- nS.

(21.2)

In the following, we shall call eK the span in the positive direction of branch b, and -d, the span in the negative direction. In most cases, the spans are nonnegative, but we shall not exclude the more general case where some of them are negative. In this connection, it should be noted that composition (d) in Fig. 21.1 is valid only if the spans of the branch are nonnegative. Furthermore, we shall call the quantity 5, - 5, = -ql the separation of the entrance from the exit.

176

V. Linear Network-Flow Problems

21.2. Maximum-Separation Minimum-Route Theorem Our arguments in this section proceed in a way perfectly dual to those in $20.2. In brief, the problem is to obtain the maximum value of -el such that when the reference branch is the voltage source of value el (i.e., when -ql = - e l ) , then there is a feasible tension configuration n

q=

q, b, over the entire network. (The problem of obtaining the mini,= 1

mum value can be reduced to that of obtaining the maximum value by reorienting the reference branch b, , i.e., by exchanging the roles of the entrance and the exit node; see Fig. 21.2.) The condition for the existence Internal branches

/

/

‘

I

I

I

/

Fig. 21.2. Minimum-route problem.

of a feasible tension configuration was established in Theorem 16.5. Since the number of elementary loops is finite, we can write out the relations (16.18) for all the elementary loops 6.Let us assume that (16.18) holds for every elementary loop 6 such that car 6 $ b, ;for otherwise, we can have no feasible tension configuration whatever value el may take. This assumption means the condition that: the sum of the spans eK and the -dK’s of the branches around every elementary closed path (we take the e, if branch b, lies on the path in the positive direction and the - d, if b, lies on it in the negative direction) should be nonnegative, which condition we shall hereafter call the nonnegative loop-span condition. The remaining relations in ( I 6.18) are concerned with the elementary loops corresponding to the elementary closed paths containing the reference

177

21. Minimum Route n

branch b,. For such an elementary loop or

6 = 1 Fb,,

where 5' = - 1

K= 1

+ 1, the relation (16.18) is of the form

where it is taken into account that (21.4) (21.3) can be summarized into

q',((")

-el (= -dl) 5 min 5

K=2

(5 = b,

+

("b, : loop)

(21.5)

K=?

to give a maximum for - e l . As is easily seen, an elementary closed path containing the reference branch b, in the positive direction consists of the b, and those branches which together form an elementary path from ns to nt . A path of this kind n

will be called a route from n, to n, . The quantity

1Q K ( t Kin) (21.5) is the K=2

sum of the e,'s of the branches contained in the route in the positive direction and of the - dK'sof those contained in it in the negative direction and will be called the distance from ns to n, along the route. In this way, we have seen that (21.5) yields a theorem that is the dual of Theorem 20.1 and is equally important though less famous. Theorem 21.1 (Maximum-separation minimum-route theorem). In a two-terminal network whose internal branches are provided with the spans e, and -d, in the respective directions, the maximum separation of the entrance from the exit is equal to the minimum distance along the routes from the entrance to the exit.

An illustration of this theorem is given in Fig. 21.3. As the reader has already noticed, if we consider a network with roads corresponding to branches, the eKand the - d , of a branch to the length or time required to pass the corresponding road, respectively, in the positive and the negative directions, then a route of the minimum distance

178

V. Linear Network-Flow Problems

i

Closed path

I

!

A.

I

(Distance of the minimum route

I

= 7)

(Maximum separation = 7; . d : tense branches)

Fig. 21.3. Illustration of the maximum-separation minimum-route theorem: Jz [-1,81, J 3 = [-1,21, Jq = [-3,4], 55 = 1-3, 11, Js = [-2,7].

=

(which may simply be called a minimum route) is a shortest route from the location corresponding to ns to the location nt . Thus, the problem we are now considering is an extension of the shortest-route finding problem. As for the relation between Theorem 16.5 in Chapter IV and the expositions in the present section, remarks similar to those given at the end of 920.2 hold. 21.3. Solution Algorithms Roughly speaking, we have two kinds of algorithms to solve the minimum-route problem. One is to start from a known feasible tension configuration q and then to increase -ql step by step in a manner dual to 0

that in 920.3.1. The other can directly find a solution if it exists or verify the nonexistence of the solution. We shall further discuss them in 421.3.1 and 421.3.2, respectively. Before entering into the details of the algorithms, we shall introduce a few technical terms. A branch b, is said to be tense in the positive direction

179

21. Minimum Route

or to be in state T + if the tension qK across it is equal to the span e, in the positive direction. It is said to be tense in the negative direction or to be in state T - if - q K is equal to the span -d, in the negative direction. Otherwise, i.e., if d, < q, < eK, it is said to be untense or to be in state T o (cf. Fig. 21.4). The tenseness is the dual concept of the saturatedness. If d, 5 q K 5 e,, then the q, is said to satisfy the span restriction. .When q K satisfies the span restriction, the quantities e, - q, and q K - d, are called the residual span in the positive direction and the residual span in the negative direction, respectively, in regard to the q K . Moreover, it is convenient to define a branch in state T + , T - , or To,respectively, as being conductive in the positive direction, conductive in the negative direction, or nonconductive.

7

T+

Fig. 21.4. Tenseness states of a branch.

21.3.1. ALGORITHM FOR THE SPECIAL CASE The algorithm to be expounded here is the dual of that given in 520.3.1, so that a brief explanation will suffice. For a tension configuration n

q = C q, b, that satisfies the span restrictions for all i

K = 2,

..., n,

we

~ = l i

search for a conductive path from the entrance to the exit (see the labyrinth algorithms in $20.3.1). If no such path is found, there is a dissection ('ill,,8,)of 'ill such that n, E 92, and nr E 'ill2, such that the corresponding i

i

i

i

segregation contains the reference branch b, in the negative direction and such that each branch contained in the corresponding cut (i.e., that segregation minus b,) in the positive or negative direction is not conductive (i.e., is not tense) in the positive or negative direction, respectively. Hence, in this case, the branches contained in the cut have positive residual spans in the respective directions. Let the minimum of those residual spans be 0

180

V. Linear Network-Flow Problems n

and define q =

qKbKby

~ = l i + l

i+l

qK = q,

+9

if b, is contained in the cut in the positive direction;

= q,

-0

if b, is contained in the cut in the negative direction;

i+l

i

i

(21.6)

otherwise, i.e., if b, is not contained in the cut.

=q K i

Then the q is a tension configuration satisfying the span restrictions for i+ 1

rc = 2, . . ., n. (The span restrictions are satisfied due to the definition of 9, and q - q is equal to the elementary cutset corresponding to the aforei+l

6

mentioned segregation multiplied by 0.) Here we have -

Vl i+ 1

’

(21.7)

-v1.

i

If a conductive path is found with regard to q, then it gives the maximum i

separation - q l , For, if there exists another tension configuration that -91

’

9 such (21.8)

-r19 i

then, in the equisignum decomposition (see Appendix 2) of Aq = fi - q, i

A1

= Aq

+ Aq +

* * *

,

2

1

some Aq will have -Aql > 0 ; therefore, Aq would be a multiple of an i

i

j

elementary cutset containing b, in the negative direction. Since it follows from the definition of the equisignum decomposition that

I 4 K I =14,-qKI

2 IAqKI, i

we have q, = 4, i

- Aq, 5 qK - Aq, < eK .i

q, = 9, - Aq, 2 9, - Aq, > d, i

i

if Aq, > 0, i

if Aq, < 0. i

(21.9)

181

21. Minimum Route

(21.9) means that in the cut corresponding to the afore-mentioned elementary cutset, a branch contained in the positive or negative direction is not conductive in the respective direction; however, this last proposition contradicts the existence of the conductive path. It is further noted that along the conductive path, each qK = eK or d,, respectively, and their sum (with appropriate signs) is equal to -q,, so that the conductive path we have i

found is a route with the minimum distance. REMARK 21.1. If e, 2 0 2 d, for all K ( = 2 , . ..,n), then we may put simplyq = 0. Hence, in this case, the preceding algorithm gives a complete 0

method of solution. REMARK 21.2. The computational labor required to obtain q from q i+ 1

i

or to know that q gives a minimum route is at most proportional to the I

number of branches since we have to perform a labyrinth algorithm and to compute 0 (cf. Remark 20.5 in 320.3.1). From the algorithmic viewpoint, there is a remarkable difference between the maximum-flow problem and the minimum-route problem, for, unlike the case of maximum flows, we need no antizigzag rule for minimum routes. In fact, as is readily seen, the dissections (’illl, ‘ill2),which we have according to the preceding algorithm, i

i

satisfy the following property: %5 i

3 2

i

i+l

23 2 ,

(2 1.10)

i+l

so that we must attain a solution q for an r less than the total number of r

nodes. Therefore, the computational labor necessary for solving a minimum-route problem is at most proportional to the product of the number of branches and the number of nodes. REMARK 21.3. The important remark of the same kind as we have made for the maximum-flow problem (Remark 20.7) holds equally well for the minimum-route problem. According to the foregoing algorithm, all the quantities we have to deal with are obtained by successive application of the operations of addition and subtraction to qK7s,e,’s, and d,’s. There0

fore, if the qK’s, e,’s, and dK’s are integers, rational numbers, algebraic 0

numbers, etc., then there is a solution (if there is a solution at all) consisting of integers, rational numbers, algebraic numbers, etc. As will be seen in 321.3.2, we can a fortiori assert that a solution is obtained by additions and subtractions of eK’s and dK’salone.

182

V. Linear Network-Flow Problems

21.3.2. ALGORITHM FOR THE GENERAL CASE By making full use of the concepts of node potentials, we can devise an algorithm by which we can solve the problem of obtaining a tension configuration in which the potential at each node has the highest possible value with the potential at the exit fixed to a given value (say 0). This problem is an extension of the minimum-route problem since its solution gives at the same time the maximum separations of all the nodes from the exit and the routes of the minimum distances from all the nodes to the exit. The algorithm is as follows, where C, denotes the potential at node n u . (i)

Initialization: Set

(, = O

la = co

and

(ii)

for

# t.

CL

(21.11)

0

0

Determine

la from C, by i

i+ 1

for every a, where nu = {h,,b,

. . .},

npi = a-bKi,

6 - n u = {b,,,

biz,.

. .}, (21.13)

nyi = a+bAi.

(iii) If for some r (less than the total number of nodes), we have

5,

r+l

= (,

for all CL,

r

then the tension configuration determined by [, = la is a solution. The r

maximum separation is given by [, - p,, and the minimum route can be found by performing a labyrinth algorithm for obtaining a conductive path from ns to n, with regard to the tension configuration thus obtained. (iv) If i, < i,for some a and some i greater than or equal to the i f 1

i

number of nodes, then we assert the nonexistence of solution. REMARK 21.4. The foregoing algorithm can be expressed in terms of matrices and vectors. We define matrix 0 = (Sap) by

9,"

=0

9,"

= mince,,

for all a,

, e , , , . . . ; - d L l , -da2, . . .], where db,, = nu - np and db,, = np - n u ,

= 00

if there is no branch connecting nu and n4,

(21.14)

183

21. Minimum Route

and calculate the vectors

6 = (Ca) by i

5

i+ 1

i

ra = min(9,8 + Cs)

=0* 5 :

i+l

i

(21.15)

i

I9

starting from

The operation (*) in (21.15) may be regarded as a kind of matrix multiplication where we use the operation of taking the minimum and the addition in place of ordinary addition and multiplication, respectively. For example, we may write i times

5 =(0* i

1 . .

* 0)* G .

(21.16)

0

[The interested reader may investigate the properties of the algebraic structure whose elements are real numbers and the symbol co, and in which two operations “ ” and “min” are defined. Introduce the matrix calculus over this algebraic structure and note that many formulas in ordinary matrix calculus remain to hold mutatis mutandis. In particular, consider the meaning of the powers of 0 with regard to “ * ”.I

+

REMARK 21.5. In the preceding exposition of the algorithm, we did not give a strict definition of the roles of the symbol co. Of course, we have to define co + x = x + c o = c o , min(x,

00)

= x,

min(co, co) = 00

+ co = co,

(21.17)

where x is an arbitrary real number. In this connection, it should be noted that although some dK’smay be - co and some ex’s may be co, we never encounter -co in the process of calculation according to the above algorithm. In the strict sense, the above algorithm works only when there is no pa equal to co. If some %.)sare co, we cannot determine definitely the tensions across the branches whose endnodes have infinite potentials. Nor can we detect a closed path around which the nonnegative loop-span condition fails to hold if the potentials at the nodes around it are given infinite values. Thus, in such a case, we can conclude only that if pa = 00, then there is a cut separating nu from n, in which the branches contained have infinite spans in the respective directions. Sometimes, however, this kind of conclusion is sufficient (e.g., for the application to the problems treated in $23).

184

V. Linear Network-Flow Problems

It is possible to modify the foregoing algorithm so as to be valid for the case where I, = co occurs. (The proof of its validity, which can be done in the same way as we shall d o below for the original form of the algorithm, is left to the reader, where it should be noted that the network under consideration is assumed to be connected.) For that purpose, we may deal with the linear forms xw + y instead of simple real numbers, where x and y are real numbers and o is a “very large quantity,” i.e., we assume

+(xo +y ) = min(x,w + y l , x2 w + y 2 ) 00

= x,o

00,

+y,

=x,o+y,

+ min(y,, y 2 ) min(co, x o + y ) = xw + y , = xco

co + 00 = 00, if x1 < x2,

(21.18)

if x, > x 2 , if x1 = x2 = x, min(co, co) = CO.

+

We simply write y , o,xw, etc., for Ow + y , lo 0, x o + 0, etc., respectively. In (21.12) or (21.14), we put eK = o or dK = -o wherever we should usually put eK= co or dK = - 00. I n contrast therewith, we put 5, = 0 and 0

[, = a, for all a ( # t ) at the initial stage. At the final stage of the compu0

tation of a solution, we may substitute a sufficiently large real number for o in the expression of every relevant quantity (simultaneously for all the occurrences of o). REMARK21.6. The computational labor required for soving a problem according to the algorithm in this section is estimated as follows. The labor to calculate ( from 4 by (21.12) is proportional to the number of i+ 1

I

branches. Since we reach a solution (or know the nonexistence of solution) before ( is calculated, where M is the number of nodes, the labor to solve a M

minimum-route problem is at most proportional to the product of the number of branches and that of nodes. It is worthwhile to note that this amount is the same in magnitude as the amount of labor estimated in Remark 21.2 with regard to the algorithm for the special case in 521.3.1. Let us now turn to the proof of the validity of the algorithm given previously, assuming the potentials [, are all finite.

185

21. Minimum Route

First, the [ i s are shown to satisfy the span restrictions for all the branches. In fact, for some r, we have

Pa = 5, = 5, r+l

for all u,

r

and, for an arbitrary branch b, such that db, = na - np, (21.12) implies the relations

or Second, it is to show that if the 5,’s are a set of node potentials which satisfy the span restrictions with [, = 0, then 5, 5 [, for all a. To show this, let us note that by definition,

5, 2 5,

for all a,

(21.20)

for all a,

(21.21)

0

and that if

5, 2 5 a i

then for a branch such that db, = n, relations (a

5 5 p + e, 6 5 p + e, ,

- np, the span restriction imposes the l p

i

5 C,

- d,

S 5, - d, .

(21.22)

i

(21.21) and (21.22), together with (21.12), yield

5,

s i +r,

for all u.

(21.23)

1

Therefore, by induction starting from (21.20), we have

5, s 5,i

for all u and all i,

so that we have Pa

2 5,

for all a.

Next, let us consider the meaning of 5,. We shall prove that i

(21.24)

5, is equal i

to the minimum of the sums of the spans (in the respective directions) of the branches along the (not necessarily elementary) paths of lengths not exceeding i from na to nt (or equal to co if there is no such path). This proposition holds for i = 0 and i = 1, as is evident from the definition.

186

V. Linear Network-Flow Problems

Let us assume that it holds for an i and consider the paths of lengths not exceeding i + 1. Each of those paths is either of length not exceeding i or consists of a branch b, for which d'b, = nu (or 8-b, = nu), and of a path of length not exceeding i from the other endnode of b, i.e., from F b , E nB (or d'b, 5 ns), to n,. This fact and the relation (21.12) prove the validity of the proposition for i + 1. In particular, pa is equal to the minimum of the sums of the spans of the branches along all the paths from na to n,. Finally, we have to consider the convergence of the sequence (Ca, [,, 0

1

[, , . . .). As we have just shown, the sequence is monotone nonincreasing 2

(beause the set of paths of lengths not exceeding i + 1 includes the set of paths of lengths not exceeding i ) . If r is greater than or equal to the number of nodes M, a path of length r contains necessarily a closed path. Therefore, if the nonnegative loop-span condition holds, the sum of the spans along a path of length r is not smaller than that along a path of shorter length, i.e., the path obtained by removing all the closed paths contained in it. This means that, for any c1 and an r greater than or equal to M , there is a number r' ( < r ) such that [, 2 [, . Taking account of the monotone nonr

increasing property of

r'

la with regard to i, we can conclude that for any a I

and an r 2 M, there is an r' < M such that [, = [, . Hence, there is an r

r'

r (<M) such that [a = [a = [a r

r+l

=

-

-

a

for all u,

r+2

if the nonnegative loop-span condition holds. Let us then consider the case where the condition fails, where there is a closed path along which the sum of the spans is negative. In that case, we can find from one of the nodes along the closed path, say nu, to n, ,a path that contains, i.e., passes, the closed path as many times as we want, so that we can find a path along which the sum of the spans is as small (i.e., negatively large) as we want. Therefore, the sequence (5,) decreases infinitely as i increases. i

REMARK 21.7. It should be noted that in the foregoing proof of the validity of the algorithm, we have also given a direct proof of Theorem 21.1 as well as a constructive proof of the tension part of Theorem 16.5, i.e., the sufficiency of the nonnegative loop-span condition for the existence of a feasible tension configuration. Furthermore, we note that if the nonnegative loop-span condition holds, [,is equal to 0 for every i. i

187

21. Minimum Route

EXAMPLE.Let us solve the problem illustrated in Fig. 21.3 of 521.2. For that network, the matrix 0 defined in (21.14) is of the form shown in (21.25), where s = 1 and t = 4:

@=

1 2 3 4

0

8

1 1

0

t

o

c = (a,

00, CO,O),

2 3

4

0

3

2

(21.25)

to.

1 7 0

5 = (9, 1,5,0>,

0

2

51 = (CO, 1,7,0),

5 = (7, 1, 5,O) = 5 3

4

c.

(21.26)

From (21.26), we see that the maximum separation of ns from n t , as well as the distance from ns to n, along a minimum route, is equal to = 7. In Fig. 21.5, the branches for which

%,

Fig. 21.5. Minimum routes to n4.

4,

A

Ca

-

- l~= e,

(ab, = na - nS)

(21.27.1)

na - nS)

(21.27.2)

or

4,

G

pa - tS= -dK

(db,

=

are drawn by bold-line arrows. [A branch for which (21.27.2) holds is given an arrowhead opposite to that indicating its proper orientation, although there is no such branch in this case.] These branches form minimum routes from nodes nl, n2, n3 to n4.

REMARK21.8. When computing the node potentials according to (21.12) or (21.15), we must remember the values of the [:s until all the i

188

V. Linear Network-Flow Problems

(,’s are determined. It is possible to apply the alternative method accordif1

ing to which we discard the old value i,as soon as a new value Ca is determined, making use of the

5,

i

5, in

instead of the

i+ I

i+ 1

the subsequent

i

calculation of (21.12) or (21.15). This latter method is sometimes called the successioe replacement to distinguish it from the original method, called the siniultaiieous replaceinent. In general, we can attain faster convergence by the successive replacement than by the simultaneous, especially when the nodes near the exit are scanned earlier and those near the entrance later in calculation of (21.12) or (21.15). It is easy to see that the %s: obtained by the successive replacement are the same as those obtained by the simultaneous replacemenr. (The interested reader may make comparison of the relations between these two methods with those between the Jacobi iteration and the Gauss-Seidel iteration method for solving a system of linear equations.)

REMARK21.9. The essential part of the foregoing algorithm is to classify the paths from norto n t of lengths not exceeding i 1 with respect to the node next to na [i.e., the node na, or ny, in (21.12) of (ii)], and then to determine the minimum over all these paths as the minimum of the minima over the respective classes-the principle of optimality in dynamic programming (see Bellman [l] and [2]). Evidently, under the nonnegative loop-span condition, we may assume that a minimum route from ns to n t is an elementary path, i.e., it does not pass a node more than one time. Therefore, if we are concerned with the paths from ns to n,, we may exclude those paths from our consideration which contain the ns as an intermediate node. This corresponds to excluding from the right-hand side of (21.12) [or (21.191 those terms which contain CS, i.e., the terms for

+

i

which / l j or y j = s in (21.12) [or p = s in (21.15)]. This simplified algorithm is useful for the application to the two-terminal characteristic problem to be treated in $23.3. 21.4. Triangular Inequalities

A network &” consisting of branches with branch characteristics of the kind shown in Fig. 21.1 may be regarded as different two-terminal networks by specifying i n direrent ways the entrance and the exit nodes, i.e., by connecting the reference branch between different node pairs. Let us denote by egathe distance along a minimum route from na to np of M , i.e., the maximum separation for the two-terminal network

21. Minimum Route

189

derived from JV by specifying na and nb as the entrance and the exit, respectively. eaa will be called the terminal-pair distance from na to na . We usually put

eaa= 0

for all 01

(21.28)

for convenience.

Theorem 21.2. For any sequence of r nodes nu, , . .. nar (r 2 3), we have (21.29)

PROOF. Let us denote by Pia minimum route from nai- I to nai . A path obtained by concatenating P, , .. . , P, in this order is a route from nal to n, along which the sum of the spans of the branches is equal to the Q.E.D. right-hand side of (21.29). In a special case r = 3, we have e m 3 a1

5 ea3 a2 + ea2 , a1

(21.30)

which is an extension of the famous triangular inequality for distances. (21.29) can be obtained by successive applications of (21.30). Theorem 21.2 is concerned with the case where the reference branches form a closed path, all of them excepting one lying in the same direction. The triangular inequality for the case where the reference branches form a segregation becomes as follows.

Theorem21.3. Let us consider a network JV in which branches

b, , .. . , bKpform a segregation such that

(21.31) is a cutset and of which the branches other than bKl , ..., bKrare provided with the characteristics of the kind shown in Fig. 21.1 (see Fig. 20.1 1). Let N ibe the two-terminal network derived from N by short-circuiting b, , .. . b K i - ,, b K i - , ,. . ., b K r , with bKias the reference branch and with gKias the maximum separation or the distance along a minimum route for N i(see Fig. 21.6). Then we have

gK12 min(g,,

9

*.

9

SIC,).

(21.32)

190

V. Linear Network-Flow Problems

(a)

(b)

Fig. 21.6. Illustration of a triangular inequality.

PROOF.By virtue of the definition of g,, , there is in N 1a closed path containing b, in the positive direction around which the sum of the spans of the branches (except for bKl) is equal to g,, . Corresponding to this closed path in N , , a closed path in N can be found (cf. 99.5 of Chapter 111). Let the loop in JV corresponding to the latter closed path be (21.33) The cutset (21.31) is orthogonal to the loop (21.33) so that we have

c t"i

= 1.

i=2

Therefore, at least one of the SKi's(i = 2, . . . , Y) should be positive. Since we may assume, without loss in generality, that the loop (21.33) is elementary, one of the tKi'sis equal to 1. Let t K j o = 1. Then the loop (21.33) uniquely determines a closed path in N j ocontaining bKjo in the positive direction. Around that closed path in N j o ,the sum of the spans of the branches (except for bKjo)is evidently equal to g K ,. Hence, we have g K n

2

gKjo'

It is obvious that (21.32) follows from (21.34).

(21.34) Q.E.D.

REMARK21.10. Theorem 21.2 is the dual of Theorem 20.3, and Theorem 21.3 is the dual of Theorem 20.2. 21.5. Applications As was already mentioned, the minimum-route problem of this section has an obvious application to the shortest-path problem in a road network, about which no further explanations will be needed.

21. Minimum Route

191

The other typical application is the project scheduling problem. A branch of a network represents a job in a project. A node represents the event that all the jobs corresponding to the branches negatively incident to it have finished, and all the jobs corresponding to the positively incident branches are to start. (A network of this kind is called an arrow diagram.) We interpret the potential at a node as the time (instant) of the corresponding event and the tension across a branch as the negative of the time (duration) assigned to the correspondingjob. Then the relation of the tension across a branch to the potentials at its endnodes is exactly the relation of a job time to the time of its start and that of its finish. Let us consider a project consisting of a certain number of jobs among which an order relation (exactly, a partial order) is defined. The network, i.e., the arrow diagram (Fig. 21.7), representing the partial order relation among

Fig. 21.7. Arrow diagram of a project.

the jobs of the project involves no closed path having all of its branches contained in the positive direction. (Conversely, such a network can define a partial order among its branches.) We regard the node corresponding to the event of the start of the entire project as the entrance and that corresponding to the event of its completion as the exit, thus making the network a two-terminal network. Furthermore, we require that each job be done between a prescribed pair of times, i.e., that the job time -qK for job K should satisfy - d , L -qK2 - e K , (21.35) where d, and eK are given (d, 5 eK 5 0). Here arises the question: Is it possible to schedule the project, i.e., to assign times -qK to the jobs such that they may satisfy (21.35), and, if it is possible, what is the optimum schedule that makes the project time (i.e., the completion time of the project minus the start time of the project) shortest? As will readily be seen, this question is a special case of the minimum-route problem, where the project time corresponds to the negative of the separation of the entrance from the exit. If all the dK’sare - as is usually the case, then

192

V. Linear Network-Flow Problems

there is a solution because every closed path contains a branch in the negative direction so that the sum of the spans around it is equal to a, i.e., the nonnegative loop-span condition holds. The maximum separation (=the minimum distance) is of a negative finite value if there is a path from the entrance to the exit with all the branches contained in the positive direction and if the e,’s are finite (as is usually the case). In an optimum schedule, the job (i.e., the branches) along a minimum route from the entrance t o the exit are assigned the shortest possible times -eK so that these jobs are not allowed to be delayed. A minimum route, in this sense, is called a criticalpath. 22. RELATIONS BETWEEN THE MAXIMUM-FLOW PROBLEM AND THE MINIMUM-ROUTE PROBLEM

Comparing the solution algorithm for the maximum-flow problem in $20.3.1 with that for the minimum-route problem in $21.3.1, the reader will find a remarkable similarity between them. In either case, branches (directions taken into account) are classified into the conductive and the nonconductive, and a conductive path from the entrance to the exit or a nonconductive cut separating the entrance from the exit is sought. This is a natural consequence of the duality of the two problems, since a path is the dual concept of a cut (i.e., if the reference branch is added, the former becomes a closed path and the latter a segregation). Furthermore, both in $20.3 and in 521.3, a branch might have been defined as conductive in the positive direction, conductive in the negative direction, or nonconductive according to whether the point representing the flow or tension of the branch is on a horizontal segment of the characteristic curve but not at its right end, or is on a horizontal segment but not at its left end, or is an interior point of a vertical segment. Thus, if we have two networks with the same graphical structure and we have to solve the maximum-flow problem of one and the minimum-route problem of the other, and if in regard to a flow configuration on one and a tension configuration on the other, the conductivity of a branch of one network is the same as that of the corresponding branch of the other network, then a conductive path or a nonconductive cut can be found for both networks by one calculation. This fact will be made use of in $23. If the graph of the network is planar, a more explicit relation can be established between the two problems. Let Jlrl and Jlr2 be two networks whose graphs are mutually dual. Suppose that we have to solve the maximum-flow problem for N l with branch b, as the reference branch and the

23. Two-Terminal Characteristics

193

minimum-route problem for N z with branch b, reoriented (i.e., - b,) as the reference branch. Furthermore, we suppose that for each internal e,] branch b, of N l ,the I" = [b", cK] is numerically equal to the J, = [aK, for the corresponding branch b, of N z . Then the vector q numerically equal to the flow configuration 5 of a solution for N l is the tension configuration of a solution for N z, and vice versa. Moreover, the set of branches forming a minimum cut in N , corresponds to the set of branches forming a minimum route in N 2 ,and vice versa. In general, the minimum-route problem is easier to solve than the maximum-flow problem, as we see from the comparison of $20.3 and $21.3. Hence, in order to solve the maximum-flow problem for a planar network N l , we may first construct a dual network N z of N , (as shown in $12) and then solve the minimum-route problem for N z . 23. TWO-TERMINAL CHARACTERISTICS

A typical form of linear network-flow problem will be treated. Not only are there a great many problems that can be formulated in this form, but also the fundamental technique is afforded to attack a more general problem in $24. We shall investigate the problem from the algorithmic standpoint in this section in contrast with the theoretical considerations in Chapter IV, $17. 23.1. Introduction The problem to be treated in this section is to determine the twoterminal characteristic of a two-terminal network whose internal branches have step-function characteristics as well as the flow and tension configurations corresponding to the points on the two-terminal characteristic curve. We shall assume throughout the following that the step-function of the characteristic of every branch consists of a finite number of steps in order to assure the finiteness of computational process. The algorithm to be developed in the following will make full use of those algorithms established in $20 and $21. Conversely, the maximumflow problem and the minimum-route problem are special cases of the problem treated in this section. In fact, the maximum-flow problem is equivalent to that of obtaining the flow configuration (in the two-terminal network N consisting of branches with characteristics of Fig. 20.1) corresponding to a point with a positive tension on the two-terminal characteristic curve for N . [From the general properties of two-terminal

194

V. Linear Network-Flow Problems

characteristics ($1 7 of Chapter 1V) and the fact that there is no voltage source in N , it follows that the two-terminal characteristic of N should also be of the kind Fig. 20.l(e). Therefore, the flow in the reference branch takes its extremum value when the tension is nonnull.] Similarly, the minimum-route problem is equivalent to that of obtaining the tension configuration (in the two-terminal network N consisting of branches with characteristics of Fig. 21.1) corresponding to a point with a positive flow on the two-terminal characteristic curve of N . An evident but important note is to be added. Let (p', -9,) be a point on the two-terminal characteristic curve of N = N ( 6 ; ref., C, , . .., C,) and let { = C tKb, and 4 = C 4, b, be the corresponding flow and tension K

K

configurations on N.Then (6, 4) is also a solution for the network obtained by replacing the reference branch b, by the current source of value n

tl. Therefore, (Z2, . . ., tn)minimizes the function cp,(Y) under the K=2 condition that the flow t' in the reference branch b, be fixed to t' where it should be noted that

(pl(e') = 0

for the current source. Similarly, I

C t,bK(qK)under

(&, . . . , 9,) minimizes the function

the condition that

,=2

the tension q, across b, be fixed at 4,.

23.2. Fundamental Theorems To begin, let us introduce some preliminary concepts. Consider the flow t" in branch b, and the tension q, across it, which together form an admissible pair, i.e., (t",q,) E C, . According to the positions of the point (tK,q,) on C, , we distinguish the four following cases (see Fig. 23.1): is an interior point of a horizontal segment of C, . is an interior point of a vertical segment of C, . is at the left end of a horizontal segment, i.e., at the upper end of a vertical segment of C, . (d) (t",q,) is at the right end of a horizontal segment, i.e., at the lower end of a vertical segment of C, . (a) (b) (c)

(t",q,) (tK,q,J (t",q,)

In each of these four cases, AtKand Aq, satisfy the conditions shown in the tabulation of (23.1) if and only if the point (Y + AY, q, + Aq,) lies on the same vertical (At" = 0) or horizontal (Aq, = 0) segment of C, as the point (t",q,) lies, where - Ab", AcK, - Ad,, Ae, are the distances from q,) to

(r,

23. Two-Terminal Characteristics

195

Fig. 23.1. Conductivityof a branch. (a) Conductivein both directions: Ab" =< Ag" =< AC", Ad, = Aq, = Ae, = 0.(b) Nonconductive:Ab" = Ag" = AC" = 0, AdK5 AqK5 Ae, . (c) Conductive in the positive direction only: 0 = Ab" 5 A F 5 AC", Ad, AqK2 Ae, = 0. (d) Conductive in the negative direction only: Ab' jA F 6 AcK= 0, 0 = Ad, 5 Av, 5 Ae"

.

the adjacent corner point of C, (cf. Fig. 23.1): (a) (b) (4 (d)

Aq, = 0 AbK5 At" 5 AcK A t K= 0 0 A t K5 AcK Ab" 5 A.5" S 0

Ag"=O Aq, = 0 Ad, 5 Aq, 5 b e , Ad, 5 Aq, 5 0

(23. I )

0 5 Aqe 5 Ae,

In case (a), (b), (c), or (d), branch b, is said to be conductive in both directions, nonconduciive, conduciive in the positive direction only, or conduciive in the negative direction only, respectively, with regard to (Y, q,).

196

V. Linear Network-Flow Problems

Next, suppose that there is a given two-terminal network JV as well as a compatible pair of flow and tension configurations (5,q). Then, we define the associated maximum-Jow problem on A'- in regard to (5, q) as the problem of finding the incremental flow configuration A4 such that the pair (5 + A t , q) is compatible and that the incremental flow A t 1 in the reference branch b, takes its maximum (or minimum) value, i.e., solving the maximum-flow problem with Ac"'s and -Ab"'s as the capacities of the branches to obtain A t . In other words, with the tension configuration fixed the problem is to maximize the entrance-to-exit (or exit-to-entrance) flow through the internal branches of JV. Similarly, we define the associated minimum-route problem on JV in regard to (5, q) as the problem of finding the incremental tension configuration Aq such that the pair (5, q + A q ) is compatible and the incremental tension - Ar], across the reference branch b, takes its maximum (or minimum) value, i.e., solving the minimumroute problem with Ae,'s and -AdK's as the spans of the branches to obtain A q . In other words, with the flow configuration fixed, the problem is to maximize the entrance-from-exit (or exit-from-entrance) separation. It is obvious that these two associated problems always have a solution (finite or infinite). We are now in the position to state the fundamental theorem:

Theorem 23.2. For a compatible pair of flow and tension configurations, the associated maximum-flow problem admits a solution with max A<' > 0 if and only if the associated minimum-route problem admits no solution with max( - A q l ) > 0. Similarly, the associated maximum-flow problem admits a solution with min A t 1 < 0 if and only if the associated minimum-route problem admits no solution with min(-Aq,) < 0. PROOF.We have only to prove the first half of the theorem since the second half is different from the first only in the orientation of the reference branch. As follows from Theorem 20. I or from the algorithm in $20.3.1, the associated maximum-flow problem admits a solution with max At' > 0 if and only if there is a conductive path from the entrance to the exit (i.e., a path in which every branch contained in the positive direction has a positive Ac" and every branch contained in the negative direction has a positive -Ab") or, equivalently, if and only if there is no nonconductive cut separating the entrance from the exit (i.e., no cut in which every branch contained in the positive direction has null Ac" and every branch contained in the negative direction has null Ab"). From Fig. 23.1 or (23.1), it is seen that the aforementioned path consists of branches (contained in the positive direction) with null AeK's and those (contained in the negative

197

23. Two-Terminal Characteristics

direction) with null Ad,'s and that the aforementioned cut consists of branches (contained in the positive direction) with positive AeK'sand those (contained in the negative direction) with positive -Ad,'s. Then, by Theorem 21.1 or the algorithm in $21.3.1, we can conclude that the existence of such a path (or the nonexistence of such a cut) is necessary and sufficient for the associated minimum-route problem to have no soluQ.E.D. tion with max( -Aql) > 0. Theorem 23.1 affords the basis for the two-terminal characteristic algorithm in $23.3 in which we solve the two kinds of associated problems alternately to determine the vertical and horizontal segments, one at a time, of the two-terminal characteristic curve. The following theorem, a special case of Theorem 17.2 (Chapter IV, $17), is useful for practical applications.

Theorem23.2. Let 6 and 6 + A k be flow configurations and q and q + Aq be tension configurations such that (<", q,) E C,, (t" + AtK,q,) E C, , and (<",qK Aq,) E C, for all K ( # I). Then we have

+

and n

n

(23.3)

<

where q = -ql, Aq = -Aq,, = < l , A t = A < ' , and q K and convex functions associated with C, .

$K

are the

23.3. Solution Algorithm The problem is to determine the two-terminal characteristic curve C of the two-terminal network N ( 6 ; ref., C 2 ,..., C,,),as well as the flow and tension configurations corresponding to the points of C . We assume that a pair of flow and tension configurations (6, q) is known such that 0 0

(<",q K )E C, for every K ( # l), from which we start the solution algorithm.

(If no such pair is known, we have to resort to the method to be developed in $24.) T o begin with, we solve the associated maximum-flow problem in regard to t6, q) to obtain a solution A6 (with At' 2 0). If A t i = 00, then 0 0

0

0

0

198

V. Linear Network-Flow Problems

C has the horizontal segment which passes

- q l ) and extends in-

((I, 0

0

finitely to the right. Otherwise, we put

5 = 5 + A5

(especially,

11=11

(especially,

1

1

5' 1

0

0

+ A0t 1 2 0tl),

= t1 0

(23.4)

- q I = -ql).

0

1

0

Obviously, the associated maximum-flow problem with regard to

(6, q) 1

1

can have no solution with A t ' > 0. In general, let ( 6 , q ) be a compatible pair of flow and tension 2i-1

2i-I

configurations for which the maximum-flow problem has no solution with A t 1 > 0. By virtue of Theorem 23.1, the associated minimum-route problem with regard to ( 6 , q ) has a solution Aq with -Aql > 0. 2i-1

If -Aql

=

2i-1

2i- 1

C has the vertical segment passing (

00,

2i- I

2i- 1

t1 2i-1

, -ql) and 2i-1

extending infinitely upwards. Otherwise, we set

5 = 2 i5- I

(especially,

+ A2 i q- I 2i-1

[especially,

q =q

2i

t 1= 2i

2i

[I), 2i-1

-qI = - q I 2i

2i-1

+ (-Aql) 2i-1

(23.5)

> -qll.

The associated minimum-route problem with regard to

2i-1

(6, q) has no 2i 2i

solution with -Aql > 0, so that the associated maximum-flow problem has a solution A6 with A t 1 > 0. If A t 1 = 00, C has the horizontal segment 2i

passing

2i

2i

-ql) and extending infinitely to the right. Otherwise, we set

((I, 2i

2i

6

=

2i+l

5 f A5 2i

(especially,

2i

(especially,

11 = q 2i+I

In regard to (

2i

6 , 2i+l

5' 2i+l

-ql 2i+ I

=

t1+ A t 1 > t'), Zi

2i

2i

(23.6)

= -ql). 2i

q ), the associated maximum-flow problem has 2i+l

no solution with A t 1 > 0. Solving alternately the associated maximum-flow and the associated minimum-route problems in this manner, we can determine the horizontal segments and the vertical segments of C, one at a time, from the lowerleft to the upper-right direction until we encounter a semi-infinite vertical or horizontal segment (cf. Fig. 23.2). The lower-left part of C may be

199

23. Two-Terminal Characteristics

obtained by computing the solutions of the associated problems with At1 < 0 and with -Aql 0 (see Fig. 23.2). Since is equal to the sum (with signs taken into account according

-=

Zi+ 1

to the directions of the branches) of the abscissas of the vertical segments of the characteristic curves of the branches which form the cut correspond-

Fig. 23.2. Two-terminal characteristic curve.

(6 , q), and 2i Zi since 5' > t', it is impossible for the sequence (tl, t1= t', 5' = ti, . ..) 2 i + l 2i-1 0 1 2 3 4

ing to the associated maximum-flow problem in regard to

to continue infinitely if the number of branches is finite and every C, has a finite number of vertical segments. Similarly, the sequence (ql = q l , 0

1

q1 = q l , . . .) cannot continue without end. Hence, for a certain finite r, 2

3

we must have either

t1 = a 2r- 1

or

-ql = a . 21

Theorem 23.3. The two-terminal characteristic curve C of a twoterminal network JV consisting of a finite number of branches, each of

200

V. Linear Network-Flow Problems

whose characteristic curves C, is a step-function with a finite number of corner points, is also a step-function with a finite number of corner points.

REMARK 23.1. Instead of beginning the solution algorithm with the maximum-flow problem with regard to (6, q), we may begin it with the 0

0

minimum-route problem with regard to the same (6,q). No essential 0

0

alteration occurs in the subsequent steps.

REMARK 23.2. As is seen from Fig. 23.1 or (23.I), the path (minimum route) which we find when we get a solution of the associated minimumroute problem with regard to ( 6 , q ) is conductive with regard to 2i-1

2i-1

(6 , q) so that in solving the associated maximum-flow problem with regard 2 i 2i to ( 6 , q), it may be used as the first conductive path along which an in2i

2i

cremental flow is assigned. Similarly, the (minimum) cut which we find when we get a solution of the associated maximum-flow problem with regard to ( 6 , q) is nonconductive with regard to ( 6 , q ) so that in Zi

2i

Zi+l 2i+l

solving the next associated minimum-route problem with regard to ( 6 , q ), it may be used as the first nonconductive cut across which the Zi+l 2 i + l

tensions of the branches are raised (or lowered). Hence, if we solve the associated problems by the algorithms of $20.3.1 and $21.3.1, we can search for a conductive path or a nonconductive cut according to the labyrinth algorithm in $20.3.1without considering which of the associated problems is being solved. We may alter the flows if a conductive path is found, whereas we alter the tensions if a nonconductive cut is found. In this way, the two kinds of associated problems play perfectly equal roles. This is a characteristic feature of the method widely associated with the names of Ford and Fulkerson [I]. In contrast with this, when we solve the associated minimum-route problem in regard to ( 6 , q ) by the algorithm of $21.3.2,we can disZi-1

Zi-1

pense with the values of q to determine the new q . [By virtue of this ~~

~

Zi

Zi-1

fact, we can start the solution algorithm with the associated minimum-route problem (see Remark 23.1) i f w e know only a flow configuration 6 for 0

which a partner tension configuration q exists although q is not numerically 0

0

known.] In fact, let e", and d", denote the ordinates of the corner points of the vertical segment with abscissa 5" of C,, and let [, , A[,, and 2i-1

c,

20 1

23. Two-Terminal Characteristics

denote the node potentials corresponding respectively to q , A q , and 2i-1

2i-1

9,then we have 2;

E, = qti 2i- 1

a, = 2i-1 ilti + Adti.

+ be,,

5" , qK ) is an interior point of a horizontal segment of 2i-1 set E, = d, = qti .] Since (21.12) takes the form (23.8) for A l Z , [If (

(23.7) C,, we

2i-1

2i-1

AC, = min[AC,; i- 1

ACp,

i

i

+ Ae,, , . ..;

A[,, - A d A , ,. . .],

(23.8)

1

and since

-

=rp,

ACYj - Addj =

+ E K j - r,

r,, -

dd,

-

r,

9

9

we can make use of the iterative formula [obtained by adding 5, to (23.8)] Ct

=O,

4'a 0

0

C,

i+ 1

= minCt ; i

=

i

(a Z t ) ,

+ ZKI . . . ;

Cut - ddl,...I,

(23.10)

i

r, = C, m

to determine the la's. [Similarly for (21.15).] (23.10) can be obtained also by recalling that for a given 5" , we have ( 5", q,) E C, if and only if 2i-1

2i- 1

d;, 5 q, 5 2,. Furthermore, in the next associated maximum-flow problem with regard to the ( 5 , q) thus determined, the search for a conductive path 2i 2i

is in general easier and more effective to perform than it is with regard to the ( 5 , q) determined by the algorithm in 921.3.1; i.e., starting from the 2i 2i

entrance node we can go through the network straight to the exit, seldom examining at a node na the worthless possibility of a conductive branch incident to na from whose other endnode there is no conductive path to

202

V. Linear Network-Flow Problems

the exit. I n fact, it will readily be seen from the way of determining the la’s in $21.3.2 [or (23. lo)] that if there is a conductive path from the entrance to a node n u , then there is necessarily a conductive path from nu to the exit. I t should be noted that whichever algorithm we may use, each of the new node potentials is not less than the corresponding one in the previous stages. REMARK 23.3. From the preceding algorithm, it can be concluded that the coordinates of a corner point of the two-terminal characteristic curve C are determined by repeated applications of addition and subtraction to tK’s,qK’s, and the coordinates of the corner points of the C, character0

0

istic curves of the branches. Thus, if the coordinates of the corner points of the C,’s as well as the t K ’ s and qK’s are integers, rational numbers, or 0

0

algebraic numbers, so are the coordinates of the corner points of C. Moreover, a flow configuration and a tension configuration corresponding to a corner point of C can be determined in the same manner. Actually,

(b)

b“

d.

-1 -1

-1

1

1

-2

-I

-1

2 I

2 1

-1 -1 -1 -1

-2 -2

1 2 4

1 1 1

-4

2

2

-I

(4

Fig. 23.3. A network and its branch characteristics.

203

23. Two-Terminal Characteristics

we can show a fortiori that the tK'sand q,'s may be assumed to be derived 0

0

from the coordinates of the corner points of the C,'s by additions and subtractions (see 824.1). These facts are generalizations of those noted in Remarks 20.7 and 21.3.

EXAMPLE.We shall compute the two-terminal characteristic curve C of the two-terminal network Fig. 23.3(a) where each branch b, (K = 2, . ., 7) has the characteristic curve C, of the form shown in Fig. 23.3(b), and the ordinates e, and d, of the horizontal segments as well as the abscissas cK and b" of the semi-infinite vertical segments are shown in Fig. 23.3(c). (The middle vertical segment lies on the q axis.) Obviously, we can start from the flow and tension configurations (5, q) = (0,O). In

.

0

0

the following, we shall indicate the ith associated maximum-flow problem by ( F i > and the ith associated minimum-route problem by ( R i ) . We shall make use of abbreviations 7

I

( F l ) : Since only 5" = 0 is compatible with q K = 0 for every IC, the first associated maximum-flow problem has the solution A t = 0 (At1 = 0) 0

0

so that we have

(Rl): The 2,'s and 2,'s have the same value as the e,'s and d,'s so that we have the matrix 0 in (23.12) [cf. (21.14)]:

@=

1 2 3

0

1 2 0

4 5 ~

0 ~~

3 3

3

4

5

1 0

2 1

co 1

co. co

2

0

2

1 0 4

4

1 0

3

2

(23.12)

2 0

~

Computation according to (21.15) with the improvement suggested in Remark 21.8 (the multiplication by l, is performed to the rows 5, 4,3, 2, 1

204

V. Linear Network-Flow Problems

in this order) proceeds as in (23.13)

(23.13)

to yield

0 = (4, 3,3,2, O), q

= (-4;

(23.14)

1, I , 1,0, - 1, 3,2).

2

Thus we have

K

2

3

4

5

6

7

8

Ac"

1 0

0

1

0

0

0

2

0

0

0

1

0

0

-Ab"

(23.16)

5 = 5 + A t = (1 ; 1, 0, 1 , 0 , 0 , 0 , l), 3

2

2

A5=At1=1, 2

2

t=1,

q=q=4,

3

3

2

(23.17)

205

23. Two-Terminal Characteristics


2

3

4

5

6

I

8

h

w 1

2 -2

w

l

2

4 -2

2 2

(3,

'\I

-2

1

-1

4

3

(23.18)

5

u\

1 2 3

o=

4 5

o -1 2 co co

w

2

c 1 0 2 2

0 2

-1 CQ

o

c

(23.19)

o co.

W

1 0 -2

4

2 0

(23.20)

s=s=s, 2

1

from which we have q=(-5;1,2,2,1,-1,3,2),

(23.21)

4

and 5=5=1,

q=-q1=5,

4

4

3

Aq=q-q=l,

4

3

Y4 = Y3 + ( . Aq

m4 = O3 = 4,

3

(F3): From

4

3

(23.22)

= 1.

3

6 = 5 in (23.17) and q in (23.21), we have (23.23): 4

3

4

K

2

3

4

5

6

I

8

Ac" -Ab"

0 1

2 0

0 0

1

0 1

0 0

1 ' 1

0

(23.23)

206

V. Linear Network-Flow Problems

Thus we have

5 = 5 + A & =(2; 5

4

1, 1, l , O , -1,O, 2),

4

A(=A('=l, 4

4

(=2,

q=q=5,

5

5

m5 = Q4 + q . A< = 9, 4

(23.25)

4

Y5 = Y4 = 1.

4

( R 3 ) : Computation proceeds as follows: K

2

&

c 1

k

4

3 o

2 2

c

o

1

\ B a\

6

1

-

-2

7

1

4

C

4

3

o

-2

--a,

2

8 '

(23.26)

2

5

l 1 2 3 4 5

o -1 -2

K

2

3

4

ACK

0 1

1 1

0

- Ab"

0

@=


5

c

2

c

3

0

c c

o

o a

(23.27)

, l

o -2

4 m 0

6

7

8

1

1

0

0

1 0

0

0 -1

0

o

1

2

-1

co 0

o 0

2

0

3

: 5

0 '

(23.30)

207

23. Two-Terminal Characteristics

6 = 6 + A{ 7

@=

6

1 2 3 4 5

= (3; 1 , 2 , 1 , O ,

o -1 -2 w W

(23.3 1)

w

w

2 -1 Q)

w

1 o

0

5=(00,00,00,00,0), 0

- 1, 1,2),

6

-1

-4

w

,

W

CO

w o -2

w w 0

s=s=f. 1

(23.3 3)

0

C

Fig. 23.4. Two-terminal characteristic of the network of Fig. 23.3.

(23.34)

208

V. Linear Network-Flow Problems

\ a U\

o=

2

3

1

o

w

w

00

0 0 ,

-1 -2 co

0 2

1 o

w w o

00

5

-1

-1

a,

-v],

(23.35)

w w

0

-l,w,w-l),

Av] = V] - V] = 2 0 - 6 ,

= 20,

+ 57 . Aq

-2

-4

a,

q=(-20;0-l,w,2,ly

Y = Y7

5

2 3 4

V] =

4

l

(23.37)

7 =z

3 + 3 . (20 - 6) = 60 - 15.

(23.37) gives information about the tension configuration as well as the value of Y for sufficiently large w. More precise discussion could be carried out based directly upon (23.34) as follows. (23.34) shows that there is a nonconductive cut which dissects the set of nodes % into the set {n,, n2, n 3 , n4} of the nodes with %, = 00 and the set i n 5 } of the node with cs = 0 # 00. The cut consists of branches b7 and b, (both contained in the positive direction). Therefore, we can put Aq = p( - b,

+ b, + b8)

209

23. Two-Terminal Characteristics

for an arbitrary positive p to obtain a tension configuration q

=q

+ Aq

7

satisfying the branch characteristics with

6. Thus, we have the parametric 1

expression of the relevant quantities for q

=

-qI > 6 :

q = ( - 6 - ~ ; 1,2,2,1, -1,4+~,3+p),

q = -ql = 6 + p ,

Aq=q-q=p,

(23.38)

7

Y = Y7

+ 5 . Aq = 3 + 3p7 7

where p > 0. 23.4. Applications

The problem treated in $23 has a great many kinds of practical as well as theoretical applications, from which a few typical examples will be mentioned.

23.4.1. NETWORK TRANSPORTATION PROBLEMS In Chapter I, $1, we presented a rough sketch of the two-terminal transportation network problem. Now we know exactly the method of solving the problem of that kind with the total entrance-to-exit Commodity flow as the parameter where the cost function q, for each unit road is piecewise linear with respect to the flow in it. Here we shall again confirm the correspondence between the concepts of the transportation problem and those of the network-flow problem. A unit road of transportation of the former problem corresponds to a branch of the latter, the cost for 5" units of commodity to flow in the Kth road corresponds to ( ~ ~ ( 5 (which ") is assumed to be a convex piecewise linear function), and the nonstagnation of the commodity at a junction point of roads corresponds t o the continuity (Kh. F.) of flow. The transportation plan corresponding to the flow configuration with a certain given amount of entrance-to-exit flow is a plan that minimizes the total cost q K(5") (summation over the internal K

branches) with the given amount of commodity to be transported from the entrance to the exit. Usually, the cost connected with a road is the smallest when no commodity is transported through it, so that the curve C, passes the origin (0,O) of the (t",q,) plane. This means that we can usually take (6, q) = (0,O) as the starting configurations that correspond to the point 0

0

(0,O) on the two-terminal characteristic curve C.

210

V. Linear Network-Flow Problems

It seems necessary to make additional comments on the devices useful in practical applications. First, let us consider the problem of finding a minimum-cost transportation plan with the given amount c of total entrance-to-exit flow through a given network. In this case, we may assume a new fictitious entrance node, connect a current limiter of value c between the fictitious entrance and the real entrance of the given network, and then seek the flow configuration with the maximum flow from the fictitious entrance to the (real) exit. (If the value of the maximum flow is less than c, the original problem has no solution; see Fig. 23.5.) Tranmortation network Fictitious

---c Current limiter

Fig. 23.5. Problem with a given total flow.

Next, let us consider the problem of transporting as much commodity as possible from the entrance to the exit under the condition that the total transportation cost should not exceed a given value Q. In this case, we may start from the configurations ( 5 , q) for which the total cost @ is 0

0

equal to zero and continue the two-terminal characteristic algorithm until we have Q2i

S Q < @2i+ 1

(23.39)

for some i. The solution { is obviously given by the linear interpolation as

If @ does not exceed Q for 5 with maximum t 2i

2i

= tl, 2i

the 5 itself is a solution. 2i

We often encounter the problem of a network with many entrances and many exits, where for each no E !JIs or nB E %, , the amount s" or tfl of the flow going in or out of the network at na or nB is given. In this case, denoting the set of entrances and that of exits by !JIs and !JI, , respectively,

23. Two-Terminal Characteristics

211

we may construct an augmented network by assuming a new fictitious entrance and a new fictitious exit, and connect the fictitious entrance to each node na of 'illsby the series connection of a diode and a current limiter of value sa and each node nB of 'illt to the fictitious exit by the series connection of a diode and a current limiter of value t B (see Fig. 23.6). The solution of the original problem is given by the flow configuration on the augmented network with the maximum amount of flow from the fictitious entrance to the fictitious exit. (If the maximum amount of flow is t B , then there is no solution of the original less than sa = n,E9lr

npi9lt

problem.)

as

R*

Fig. 23.6. Multientrance multiexit problem.

The network transportation problem allows a number of different interpretations besides the foregoing one. For example, we can consider the problem of constructing a road network with a given topological structure that is capable of conveying a given amount of entrance-to-exit flow, where ( ~ ~ ( 5 "is)interpreted as the cost to construct the rcth road so as to have the capacity r". Another example is the problem of transferring empty ships. We regard the nodes of a network as the cities among which commodities are traded, i.e., imported and exported, by sea through the routes corresponding to the branches. In assigning the ships to the sea routes, there is no problem if every city imports as much as it exports, but otherwise, we are forced to consider the optimum plan of transferring the empty ships from the cities where more is imported than is exported to the cities where more is exported than is imported. Let us denote by 'illsthe set of nodes corresponding to the cities where imports exceed exports and denote by 'ill, the set of nodes corresponding to the cities where exports exceed imports. We know the cost e, for transferring an empty ship through the rcth route b, ,and assume that sa( >0) is the excess of imports over exports at the city na E 'ills and t B(>0) is the excess of exports over imports at the city ns E 'illt. Thus, the problem is

212

V. Linear Network-Flow Problems

reduced to the multientrance multiexit minimum-cost network transportation problem (such as just explained), where the flow 5" in route b, is the number of empty ships transferred through it and the objective function to be minimized is the sum of the rp,(r)'s for all the routes. Here q,(t") is defined by

23.4.2. SPECIAL KINDSOF TRANSPORTATION PROBLEMS There are a large number of practical problems represented by a network-flow problem with a special topological structure, and sometimes, the same problem has more than one different network representation. In the following, we shall show the examples illustrating these circumstances. (1) HITCHCOCK TRANSPORTATION PROBLEMS AND ASSIGNMENT PROBLEMS: We consider n? producers and n consumers of a commodity located at different places. The ath producer (a = 1, . ..,m) supplies the amount sa of the commodity and the Pth consumer (P = m + 1, ..., m + n) demands the amount t o . The maximum amount that can be transported from producer a to consumer P is cap (which may be 0 or a),and it costs as much as eorp(20) to transport the unit amount there. The problem is to find among the transportation plans which minimize the sum of the surpluses at the producers (or, equivalently, the sum of the shortages at the consumers) one plan which minimizes the total transportation cost. This problem, usually called the capacitated Hitchcock transportation problem, may be formulated mathematically as follows. The nonnegative constants sa, to, cap, and eap are given for a = 1, .. ., m and P = m + I , . . .,m + n, and the variables 5, y, tP,and r p are adopted (representing, respectively, the total amount of the commodity transported, the amount transported from the ath producer, the amount transported to the Pth consumer, and the amount transported from the ath producer to the Pth consumer). These variables are subject to the equality constraints

5"=

m+n

1 rfl

(a= 1,

..., m ) ,

(P=

+ I , ..., m + n),

p=m+ 1 rn

=

1y p

m

a= I

c

m

m+n

a=l

p=m+l

t;=.&y=

(23.42)

5"Z$fl a,p

which express the continuity of the commodity flow, as well as to the

213

23. Two-Terminal Characteristics

inequality constraints, ( a = 1 ,..., m),

saz5"zo t0 = > tfl>O =

(P=m+ 1, .. . , m+n),

(23.43)

(a= I , . ..,m ; fi = m + I , . .., m + n).

cap 2 cap>=O

Let the maximum value that 5 can take under conditions (23.42) and (23.43) be g. Then the problem is to find the set of values (5, r", t", Yp) which minimizes the total cost m

m+n

(23.44) under conditions (23.42), (23.43), and (23.45) :

5 = g.

(23.45)

A Hitchcock problem is formulated as a kind of multientrance multiexit network transportation problem as follows. I t will be obvious that constraints (23.42) are embodied by a network of the structure shown in Fig. 23.7(a) where we have m nodes {nl, .. ., nm} ( =al)corresponding to the m producers, n nodes {n,,,, . . ., nrn+,,}(=%,) corresponding to the n consumers, an entrance node no = n,, and an exit node n m + n + l= n,; m + n + 2 nodes in all (a= u %, u {n,, n,}). The set of branches 23 consists of m (additional) branches connecting n, to the nodes of illl (5" being the flows in them), n (additional) branches connecting the nodes of %, to n t (tpbeing the flows in them), and m . n (intrinsic) branches connecting the nodes of to those of % ' , (tap being the flows in them); m + m * n + n = (m + l)(n + 1) - I branches in all. [Since the network is a two-terminal network, the number of branches amounts to (m + l)(n + 1) if, in addition, the reference branch connecting n, directly to n, iscounted.] The restrictions (23.43) as well as the cost function (23.44) can be realized by providing the branches with the characteristics shown in Fig. 23.7(b). In the network .A-' thus constructed, the pair of flow and tension configurations (5, q) = (0,O) is obviously compatible so that we 0

0

can calculate the two-terminal characteristic of JV beginning from that pair. The flow configuration corresponding to the largest possible entranceto-exit flow t = ?, which is obtained by the algorithm explained in $23.3,

is a solution of our problem. If

g=

m

a=l

m+n

sa =

fp,

we have a solution

B=m+l

satisfying all the supplies and demands; whichis a solution for the problem

214

V. Linear Network-Flow Problems

Fig. 23.7. Hitchcock problem.

which has the constraint relations

sa = y,

to = t o

(or tP 5

tP)

(23.43')

instead of the first and second relations of (24.43). (The Hitchcock problem is sometimes defined in this form.) For the pairs (u,p) such that it is impossible to convey the commodity from producer a to consumer p, we may put czB= 0 (or/and eao = a). Since the structure of the network for the Hitchcock problem is highly special as just shown, the solution algorithm for it can also be much simplified. The details about this possibility of simplification will be discussed in $23.4.3. The special case of the Hitchcock problem where m = n, sa = l p = 1 (and we may put caP= 1 or cap > 1) for all a's and fl's is called the assignmetit problem. If all the calf's are finite, we obviously have [ = rn ( = n ) , r" = I , = 1 (for all CI and p) in every solution. Furthermore, by virtue is equal to 0 or 1 . of Remark 23.3, we can find a solution in which every tUo

23. Two-Terminal Characteristics

215

In such a solution, the value of is equal to the sum of the eaB’sfor which yp= 1. Since 5“ = 1, there is one and only one B for each a such that tUB = 1. Similarly, there is exactly one a for each fl such that = 1. This means that the rB, when regarded as an m x m matrix, is a permutation matrix. Hence, in terms of the e,is arranged in the form of a matrix, the problem can be formulated as one of choosing m elements out of mz eUB’s in such a way that exactly one element is chosen from each row, exactly one from each column, and the sum of the chosen e,,’s is as small as possible. In practical terminology, we may state the problem as follows. We consider m persons and the same number of jobs and assume that there is a loss e,, when person a is engaged in job p. What is the best one-to-one assignment of jobs to persons in order to ensure minimum total loss? If, instead of the losses e,, of assignment, the gains e:, are given and the maximization of the total gain is required, we may convert the problem as losses by setting into a minimization problem with the eUB’s Z = max ei,,

e,, = Z - e;,,

(23.46)

a. B

because, for any assignment yB,we have

and mZ is independent of the tap’s.An efficient computational algorithm for the assignment problem will be given in 923.4.3. To the assignment problem may be given another practical interpretation, as follows. There are m glasses nl, . .., n, and m persons ,n . . . , nZn,as well as a certain amount Q of liqueur, and it is assumed that E,, of the liqueur contained in glass n, has the unit utility for person nB (a = 1, . ..,m ;p = m 1, . ..,2m). Here arises the problem of dividing the liqueur among the glasses and then distributing them to the persons, one to each, in such a way that each person gets one of the glasses of the greatest utility for him. If we denote the amounts of liqueur in the glasses by zl, ..., z, and the utilities the persons obtain by z,+~, ..., z Z mand express the way of distributing the glasses to the persons by the permutation matrix 8 = (yfl),

+

yB= 1 =0

if glass na is assigned to person n,, otherwise,

216

V. Linear Network-Flow Problems

then the conditions to be satisfied by z,, z p , and

c

Ppare

m

a= 1

zu =

Q,

zp 2 za/cap

(a= 1, . . . ,m ; B

zp = zu/cap

if t a p > 0.

(23.47.1) =m

+ 1, ...,2m),

(23.47.2) (23.47.3)

Here it is noted that because of the homogeneity of (23.47.2) and (23.47.3) in regard to za and zp , only the ratios among the z’s are of significance so that (23.47.1) may be put out of consideration. If we set 5u

= log z a 9

5s = log zp

3

eap = log cap 9

(23.48)

the conditions (23.47.2) and (23.47.3) are written as

As will readily be seen, the relations (23.49) are exactly those to be satisfied by the tensions qalracross the intrinsic branches and the potentials 5, and u [a at the nodes of with the flows Cup in the intrinsic branches (see also Fig. 23.7), where we put cap = co for the intrinsic branches. Thus, the problem is reduced to an assignment problem where not only the flow configuration but also the tension configurations are given practical meanings. There is a strange relation between the general network transportation problems and the Hitchcock problems. Obviously, the latter problems are a very special case of the former. Strangely enough, some broad subclass of the former is proved to be equivalent to a very special subclass of the latter. The former subclass consists of the problems in which every branch has the characteristic curve C, ending with semi-infinite vertical segments at both ends and in which the flow in the reference branch is given, whereas the latter subclass consists of the uncapacitated Hitchcock problems, i.e., the Hitchcock problems in which all the cup’s are infinite. (Note here that the condition that a C, should end with semi-infinite vertical segments is practically no restriction because a C, ending with a semi-infinite horizontal

23. Two-Terminal Characteristics

217

segment at an end may be substituted for by the C,' obtained by bending the semi-infinite horizontal segment of C, vertically at a sufficiently distant place.) A network N of the former class may be assumed to consist of the branches of the structure shown in Fig. 13.12 with e nonnegative (in $13.3 of Chapter IV), since any step-function characteristic can be composed from such branches, as was illustrated in Fig. 13.14, and of the reference branch which is a current source. Let us denote by N ' the network obtained from N by removing all the current sources (including those in current limiters) where we regard the set of nodes of N'as consisting of those of N (whose set will be denoted by and of the middle points of the branches [expressed in the second form of Fig. 13.12(a)] of N (whose set will be denoted by S2). As follows from the construction of N ' , every branch of N' is either a diode or the series connection of a diode and a voltage source of a nonnegative value conto a node of a2.Furthermore, it is obvious that necting a node of finding a solution for N is equivalent to finding one for N' where the inflow at a node of 9l1is set equal to the sum of the values (signs taken account of) of the current sources connected with the corresponding node of N,and the outflow at a node of 'iR2 is set equal to the value of the current limiter (or the current source in it) contained in the corresponding branch of N [cf. the equivalence transformation of type 2.1(4) in $19, Chapter IV]. Thus, we have seen that a network transportation problem with M nodes and n branches (of the form shown in Fig. 13.12) is equivalent to an uncapacitated Hitchcock problem with M entrances and n exits. In Fig. 23.8, this equivalence is illustrated.

Fig. 23.8. Equivalence between the two-terminal problem and the Hitchcockproblem.

218

V. Linear Network-Flow Problems

( 2 ) WAREHOIJSING PROBLEM:We shall take up another example of special topological structure to illustrate that among the mutually equivalent network representations of a problem, some are intuitively natural but not suitable for theoretical considerations, while others are not natural but lead us to deeper insight. The so-called n3areliousing problem is defined as follows. We consider a sequence of 111 successive periods of time and a warehouse of capacity c for a commodity. In the ith period, we purchase x i units of the commodity, store them in the warehouse where ti-’units have been held over from the previous (i - 1)st period, sell y‘ units, and hold over ti( = ti-’+ x i - yi) units to the ( i + 1)st period. For the sake of simplicity, we assume that to= trn= 0, i.e., that the warehouse is initially empty and it should be made empty after the final period. The cost of purchasing a unit of the commodity in the ith period is di,the price of selling one unit is ei,and the cost of storing a unit during the ith period is f i . In each period the warehouse has to store, although temporarily, ti-’+ x i units (the sum of the held-over and the purchased amount) so that ti-’+ x icannot exceed the warehouse capacity c. Here the question arises: What kind of plan of purchasing and selling will maximize the total profit (i.e., the total selling price minus the total purchase cost minus the total warehouse cost) or, equivalently, will minimize the total loss which is equal to the negative of the total profit? The most straightforward network representation of the problem would be as shown in Fig. 23.9, where the two nodes with labels “entrance” and “exit” are short-circuited (i.e., they are regarded as a single node).

Fig. 23.9. Warehousing problem.

219

23. Two-Terminal Characteristics

In the network of Fig. 23.9, the continuity conditions for ti, xi, and y', (1

- y',

=

ti = t i - 1

+

xi

- yi

( i = 2,.

.., m - l),

(23.50)

O=tm-l+Xm-ym,

are to be satisfied (and there is no other constraint imposed upon them). Moreover, the function to be minimized, m

m- 1

m

~ ( x, t ,y ) =i 1 dixi + C f i t i- C eiyi, i= 1 i= 1 = 1

(23.51)

is exactly the total loss. To solve the problem, we may remove the short-circuited branch (connecting the exit directly to the entrance), regarding the network as a two-terminal network, and search for a pair of flow and tension configurations in which the potential difference between the entrance and the exit is equal to zero. Then the flow configuration thus obtained gives a solution. Since the voltage sources of values ei are placed in the direction opposite to the directions of the diodes in series, we cannot adopt (6, q) = (0,O) 0 0

as a starting pair of configurations. However, if we take for q such a 0

tension configuration that all the nodes other than the exit node have an equal potential of sufficiently large negative value, the potential of the exit being null, then obviously, all the branch characteristics are satisfied by (4, q) with 5 = 0. Therefore, we can start the two-terminal algorithm with 0 0

0

5 = 0 as the initial flow configuration (see Remark 23.2). 0

It will be shown, by applying the equivalence transformations in $19 of Chapter IV, that the warehousing problem can be treated in much simpler manner than the general two-terminal problem. We transform the network of Fig. 23.9 as follows. First, we combine the entrance and the exit into one node and remove (by short-circuiting) the diodes placed in series with the current limiters [because the diodes in which the ti's and xi's flow are sufficient to constrain the flows in the current limiters to be nonnegative; transformation of type 2.3(2) in $193. The resultant network can be drawn in the form of a fan of triangles (see Fig. 23.10). Incidentally, the representation of the form of Fig. 23.10 is convenient when we consider the problem of cyclic structure, i.e., when we admit to hold over some xm - ym)after the final mth period and regard it as amount 5" (= tm-' the hold-over to the first period (i.e., we put t1= tm x1 - y'). To the

+

+

220

V. Linear Network-Flow Problems

Fig. 23.10. Another representation of the warehousing problem.

network of Fig. 23.10, we can apply the transformation of type 1, specifically that which we illustrated in Figs. 11.4 and 11.5 ($11 of Chapter 111) to obtain the network of Fig. 23.1 1. Furthermore, the current sources in the network of Fig. 23.11 may be replaced by a single current source of value c which connects the exit directly t o the entrance [transformation of type 2.1(4) and (3)]. Thus, the original problem has been converted into that of obtaining a solution of the two-terminal problem (for the network of Fig. 23. I1 with the current sources removed) which corresponds to the total entrance-to-exit flow of value c. Every branch in the network of Fig. 23.1 1 has the characteristic curve with a downwards semi-infinite vertical segment on the q axis and a horizontal segment starting from a point on the q axis and extending infinitely to the right. Therefore, starting from the null flow configuration (i.e., all 5' = x i = y' = 0), we need solve

..

.ntrance

.. .. Fig. 23.11. Equivalent representation of the warehousing problem.

22 1

23. Two-Terminal Characteristics

the associated minimum-route problem only once. In fact, if the associated problem yields a solution with the maximum separation null, then we may assign c units of flow along the straight path from the entrance to the exit consisting of the diodes only. This corresponds to null purchase and null selling, and hence, null profit (or null cost). If the maximum separation is negative, then we may assign c units of flow along the corresponding minimum route. The maximum separation cannot be positive since there is a path consisting of the diodes alone from the entrance to the exit. Thus, we have seen that we may perform the algorithm of $23.3.2 only once to solve the warehousing problem. Actually, we know more about the problem, i.e., we can construct a solution of the warehousing problem such that each of the selling, purchasing, and holding-over amounts in a period is equal either to zero or to the capacity of the warehouse. SOLUTION ALGORITHMS FOR THE HITCHCOCK TRANSPORTATION PROBLEM AND THE ASSIGNMENT PROBLEM We shall develop the solution algorithms for the problems defined in (1) of $23.4.2. First, let us take up the Hitchcock problem where we shall use the same notations for constants and variables as before. We shall arrange in the tableau of Fig. 23.12 the constants and variables concerned, where 5, or is is the potential at node na or nD, q,,, (= 5, is the tension across the branch connecting norto n,, ,and q (= C0 - 5,n+n+l = is the difference of the potential of the exit node nrP,+,,+,from that of the entrance node no which is equal to the negative of the tension across the reference branch. Since we assume e,,, 2 0, we may adopt the starting configurations in which t = 5" = = Y P= 0, il = ia= is = rl,,, = 0 for all a (= 1, . . . ,m) and fl ( = m + 1, . . . , m / I ) . Then the solution algorithms for the associated minimum-route and maximum-flow problems on the network of Fig. 23.7(a) will be stated as follows if the obvious simplifications due to the special topological structure of the network are taken into account, where we are going to determine the values of the variables corresponding to ( 5 , q ) by modifying the values correspond-

23.4.3.

rs)

c0)

+

i+l

i+l

ing to (5, q) which are given in the tableau of Fig. 23.12. i

i

Solution of the associated minimum-route problem: We adopt the algorithm of $21.3.2, also taking account of Remark 21.9. To begin we set

c,,0 = 0 =

<, 0

co

= 00

if to - tfl> 0, if t,, - t p = 0 , for all a,

(23.52)

222

V. Linear Network-Flow Problems

\ to -

...

E

...

. ..

*

.

. .

...

...

. s= -

. ..

.

.

.

5"

\

__

.

'i

...

.

. . .

...

40

...

4m

+n

Fig. 23.12. Tableau for the Hitchcock problem.

and calculate iteratively

+ cap>>

.. ., m )

(23.53.1)

(where the inner min' is taken over those p's for which cap -

> 0) and

Ca i+l

Ca

i+ 1

= min{C,, i

= min{Cp, min i

a

min'(Cp p

i

"(iC,+ l - cap)}

(a = 1,

(p = m + 1, . . .,m + n) (23.53.2)

223

23. Two-Terminal Characteristics

(where the inner min” is taken over those CL’S for which Y p> 0). We shall have for some r

5,

r+l

= 5, (= r

pa),

cs = csr ( E c,)

for all c1 and B.

r+l

Then we replace the old values of 5, and 5, on the tableau by these p:s and tp’s, and also the old values of q and (q,, - e,,), respectively, by

where the min”’ for 9 is taken over those a’s for which sa - Y > 0. (Note that, in the above calculations, the old values of 5 and q do not concern us.) REMARK 23.4. A rough explanation of the above algorithm will be given, but the detailed proof of its validity is left to the reader as an exercise. Since the existence of a solution is assured, i.e., the nonnegative = 0 for every i loop-span condition is a priori known to hold, 5, =

rm+,,+

i

i

so that we omit writing it and set q = 5, E C0. By taking account of Remark 21.9, we can calculate 9 = after all the other [ i s and ts’s have been determined. Those 5,’s for which t B - 5, 0 are set equal to 0 because if

to

0

=-

we put them initially equal to co, we should have them modified into 0 (i.e., 5, = 0) at the next step. Furthermore, we omit in (23.53.2) reference 1

to the spans of the branches connecting the nodes of ‘9I2to the exit because of the nonnegative loop-span condition and (23.52). (Consider also the fact that the spans of those branches are 0 or co.) REMARK 23.5. The contents of the tableau of Fig. 23.12 are somewhat redundant. Indeed, if 0 < FB< cap, then we should have q,, - e,, = 5, - 5, - e,, = 0, if qas - e,, < 0, then lap= 0, and if q,, - e,, > 0, then cap - Ta = 0. The possible patterns of signs in a unit square in the interior region of the tableau are those and only those listed in Fig. 23.13.

Fig.23.13. Sign patterns of the unit squares in the tableau.

224

V. Linear Network-Flow Problems

Solution of the associated maximum-flow problem: We search for a conductive path, i.e., for a sequence of row and column indices ( a o ,/lo, al, PI, . . . , a,, f i r ) such that sao - tao >0 and la0 = V , (23.55.1) p D i - taiDi > 0 and qaiDi- eaiai= 0 ( i = 0, . . . , r), (23.55.2) 5°L'+'Bi> 0 and qai+,si- e,i+,si = 0 (i = 0, . . , r - l), (23.55.3) and tflr - t f l v > 0 (Csr = 0). (23.55.4)

.

Figuratively speaking, starting from the square (containing sao - 5"" and p )in a row a, for which (23.55.1) holds, we proceed in the vertical and the horizontal directions until we arrive at the square (containing t P r- gsr and Ear) in a column 8, for which (23.55.4) holds, where we are permitted to turn from the horizontal to the vertical direction at a square with the sign pattern of Fig. 23.14(a) and from the vertical to the horizontal direction

<" = 0 (c)

Fig. 2:3.14. Paths on the tableau.

rl

225

23. Two-Terminal Characteristics

at a square with the sign pattern of Fig. 23.14(b). If such a path or sequence is found, we compute

decrease by A t the sao - tao,tSr - tsr,(cuipi- pipi)’s, and Yi+lPi7s and increase by A t the 5, p , tBr, taisi’s,and ( c a c + I p-i t a i + l p i ) ’ s . We repeat the above process until we can no longer find such a path. REMARK 23.6. The algorithm for finding a conductive path, i.e., the labyrinth algorithm, may be described as follows in the present case (cf. $20.3.1), where we make use of m + n working spaces 6,, ..., 6,,, . (i) Initialization :

(ii) Search for a cap - Ya

p such that 6,

> 0,

= co and that

q,, - e,, = 0,

6, #

00

for an a,

and convert that 6, (= 00) into 6, = a. (iii) Search for an a such that 6, = co and that

pfl > 0,

q,, - e,,

= 0,

6, # co

for a

p,

and convert that 6, (= co) into 6, = P. (iv) Repeat (ii) and (iii) until (iva) we find a p such that 6, # co and t D - tp> 0

or (ivb) we have t S - ta = 0 for every p with 6, # co, and we can no longer decrease the number of the B’s with 6, = co or the number of the a’s with 6, = co. If (iva) is the case, we define that p as Po and define successively

until, for an r, we have

226

V. Linear Network-Flow Problems

Then the sequence (a,,

Po, .. ., a,, f i r )

defined by

is the required one. If (ivb) is the case, we conclude that a desired path does not exist. REMARK23.7. Each time we solve the associated maximum-flow or minimum-route problem, we have a new value o f t or q which is strictly greater than its old value; i.e., beginning with (5, q ) = (0, 0 ) , we have in o n

(4,

q = a), where

N N+I

The total cost @ of the solution is then calculated by N

@=

Ccr-

i=l

i

(23.58)

5 ) v 7

i-l

i

as was shown in Theorem 23.2.

REMARK23.8. If the problem is to minimize @ = conditions (23.42), (23.43’), and cap 2 eU8’sinto e& = eaD - E, - E~

(a =

1 , . . ., m ;

tUP 2 0, then

1e,, Yp under the a, P

the modification of

p = rn + 1 , . . . m + n) (23.59)

with arbitrary c,’s and cp’s leaves the solutions invariant. In fact, for arbitrary t a ” s satisfying (23.42) and (23.43’), we have

(23.60) so that the difference between @ and @’ is independent of

ep’s.It is often

227

23. Two-Terminal Characteristics

convenient to choose E,'S and 8;s as E,

= min e,,

,

E,

= min(e,,

- EJ.

(23.61)

B

This modification can be interpreted from the points of view of equivalence transformations of networks discussed in $19 of Chapter IV. The reader should investigate for himself how it can be done with Fig. 23.15 as a hint.

(C)

Fig. 23.15. Cost-reduction transformation.

EXAMPLE.Let us solve the Hitchcock problem defined by the tableau of Fig. 23.16 where we want to find a solution with 5" = s' (a = 1, . .. , 4) and 5, = t D (@= 5, ..., 10). According to Remark 23.8, we preliminarily modify the problem by replacing e,, by e:, as determined by (23.59), setting El

= 3,

E2

= 5,

E3

= 2,

Eq

= 5,

Eg

E6

= 0,

E7

= 1,

E8 = 0,

Eg

= 2,

EI0

= 0,

=o

(cf. (23.61)]. The corresponding difference in the total cost is then given by C23.60) as

a) - @'= 81.

228

V. Linear Network-Flow Problems

::": N

d

u

u

&

?'; GJl 9

i': X m ' 9

GJl

.2

&

229

23. Two-Terminal Characteristics

We begin by setting all 5 , r", tP,and Y pequal to 0 and we solve the associated minimum-route problem which yields the obvious solution with all Ca, Cp, q, and qnP equal to 0. Thus we have the tableau of Fig. 23.17.

co

co

co

6 0

)( -4

l o l o

0

Fig. 23.17. Tableau 1.

Next, we begin solving the associated maximum-flow problem on the tableau of Fig. 23.17. Since Ci = 0 = q, s1 - 5' = 3 > 0, c16 - 516 = 3 > 0, q16- e;6 = 0, and t 6 - t6 = 3 > 0, the path (ao = 1, Po = 6 ) indicated by a dotted line in Fig. 23.17 is conductive, and along it, A5 = 3 units of flow are added, i.e., we change the values of 5, 5', 516, and t6 to 3 and those of s1 - 5', -
230

V. Linear Network-Flow Problems

the values of t, t2,< 2 5 , and t5 by 3 and decrease those of s2 - t2and t 5 - c5 by the same amount, c 2 5 - t2’ remaining co even though it is decreased by 3. Furthermore, on the resulting tableau, we find the path ( x o = 2, /Io = 8) along which one unit of flow is assigned, and then the

10

*

Fig. 23.18. Tableau 2.

path ( x o (ao = 4,

= 3, Po = 7) with the incremental flow A t = 2, and finally the path fi0 = 8) with A< = 1 . The resulting flow configuration is shown

in the tableau of Fig. 23.18 together with the same tension configuration as in Fig. 23.17. We cannot find a conductive path there as easily as before. Therefore, we resort to the systematic procedure proposed in Remark 23.6.

23 1

23. Two-Terminal Characteristics

Since row 4 is the only row in which s4 - 5" > 0 and uo = 4 and

6 = (al, . ..)6,;

c4 = q = 0, we set

. ., 610) = (co,coy co,0; coy coy co, 02, co, co). In row 4, column 8 is the only column in which c " ~- 548 > 0 and q48 - ek8 = 0, so that we change 68 to 4. Since t 8 - t 8= 0, we further search 85,.

for a row u in which Y8 > 0 and qa8 - e& = 0 to find u = 2, changing 6, to 8. In row 2, the columns B satisfying c2@- t2a> 0 and q2@- e;@= 0 are columns 5 and 9, so that d5 and 6, are changed to 2. Thus we have 6 =(coy 8 , c o , O ; 2, co, co,4,2, co). Although t - 5' = 0, we have f - t9 = 1 > 0, and, therefore, we have found a conductive path, which is formally constructed as follows: 69 = 2 ,

62 = 8 ,

68 = 4 ,

(ao = 4 ,

Po = 8,

= 2,

64=0

or

p1= 9),

and which is indicated in Fig. 23.18 by a dotted line. The amount of incremental flow to be assigned along it is

A t = min(s4 - 5" = 7,

c48 c29

=

- 548 = 00, cz8 = 1, - 5 2 9 = 03, t 9 - t 9 = 1)

1,

r9

and 5, t", 5"', c28 - tZ8,529, are increased by A t = 1, whereas s4 - 5", tZ8,cZ9 - tZ9,and t 9 - t9are decreased by the same amount. It is easy to see that there is no conductive path on the tableau thus modified. Hence, we completed the associated maximum-flow problem. The resulting flow configuration is shown in the tableau of Fig. 23.19. (The tension configuration there will be calculated at the next step.) During the above process, t is increased by A t = 1 1 ,whereas q remains fixed to 0. Hence, we

c48 - t48,

0

have

a1'= A@.,' = q A( *

1

= 0.

0

The next step is to solve the associated minimum-route problem with regard to the flow configuration of Fig. 23.19. Since r a - > 0 for f l = 7 and 10, we initially set <@

232

V. Linear Network-Flow Problems

-3

At=:

-1

3

1

2

0

Fig. 23.19. Tableau 3.

The calculation according to (23.53) proceeds as follows:

51 = (2,62094; 6 , 2 , 0 , 4 , 6 , 0 > , 5 =(2,3,0,3; 3,2,0,3,3,0)=5 2

=

p,

3

and we have by means of (23.54), ti = [, = 3 and the values of qa,, - eaB shown in Fig. 23.19. On the tableau of Fig. 23.19, we search for the conductive path (indicated by a dotted line) which is (ao = 4, Po = 6, a, = 1, PI = lo),

233

23. Two-Terminal Characteristics

and along which

A t = min(,y4 - c4 = 6 , 10

c4p6 - 14,6= 4,

-

10

=

co,

t 1 y 6

t'O

= 3,

- ('0

= 2)

= 2.

If the necessary alterations are made on the tableau, we can no longer find any conductive path. Thus we have completed the associated maximumflow problem with the resulting flow configuration shown in Fig. 23.20.

1

4

1

3

0

Fig. 23.20. Tableau 4.

234

V. Linear Network-Flow Problems

Since q

=3

and A[ = 2, the increment of the cost function Q' is equal to 1

2

3 ~ 2 = 6 . The next associated minimum-route problem yields the tension configuration shown in Fig. 23.20 where the calculation of potentials is:

5 = (a, a,a,00;

00,

a, 0, a,a,a),

0

51 = (396, a,4; 6, 3 , 0 , 4 , 6 , I), 5=(3,4,1,4;4,3,0,4,4,1)=&=0, 3

2

q=

17

1

s s

c4 = 4.

235

23. Two-Terminal Characteristics

In the next associated maximum-flow problem, we find a path (ao = 4, Fig. 23.20, and nothing more. The modifications of the flows along that path give the flow configuration shown in Fig. 23.21 where we see that all the (sa - 5")'s as well as all the ( t B- ta)'s are equal to 0. Therefore, we have arrived at a solution f. (If we solved the associated minimum-route problem further, then we should obtain q = m.) The increment of the cost @' during the last step is q A t = 4 x 4 = 16.

Po = 7) indicated by a dotted line in

3

2

The corresponding total cost is

The solution algorithm for the assignment problem may be simplified further than that for the Hitchcock problem. Since we can perform the solution algorithm in such a way that every tZp is always either 0 or 1 , at most one r pis 1 for each a, and at most one tapis 1 for each p (cf. Remark 23.3), we have only to remember in which column is the nonnull tapin each row, or in which row is the nonnull Fpin each column, i.e., we make use of a vector v = ( v l , . .., vm;v , + ~ ,.. ., v Z m )to describe a flow configuration where v,

=

rP= 1 ,

v,

if Y P= 1,

vB

p if

=0

if

5" =

2m

tap= 0,

(23.62.1)

p=m+l

and v,

=a

m

=0

if tB= 1ys= 0.

(23.62.2)

a= 1

From the theoretical viewpoint, either v, (a = 1, . .., m ) or vB ( p = m + 1, . . . , 201) is redundant (i.e., v, = p if and only if vB = a), but from the practical viewpoint, it is convenient to deal with this redundant information. The entries for q,, - eUBmay be omitted because they are redundant (as was already pointed out). Hence, in solving an assignment problem, we may deal with vectors v for the flow configurations and 5 for the tension configurations, each with 2m components, besides the given cost matrix err,. We start the solution algorithm with all v, = vB = 0. Solution of the associated minimum-route problem: We set

[,,= 0

if vB = 0,

0

=a [,=a 0

if v p # O ; forall a ( = l ,

(23.63)

..., m ) ,

236

V. Linear Network-Flow Problems

is's and la’s by

and then calculate iteratively the

i

i

C,

i+ 1

= min[C,, i

min(l,

+ e,,)]

(a = 1 ,

...,m )

(23.64.1)

min(Cp

+ e,,)]

(a = 1 ,

.. . , m )

(23.64.1’)

p+v,

i

(when we set cap = 1) or

C, = min[Ii,, i

i+ I

/ 3 i

(when we set cap > 1) and by

5,

=

i+l

<,

if vs = 0 ,

I

=

i - e,,,

if v , # O

(23.64.2)

i+?

for i = 0, 1, .. . , until we have

l a = 5, r

for all a and p.

i-

and

r~

r+l

-r!{

Then we set

e,

=

5,r

P,

and

=i

r

p

(23.65)

and r] = min

v,=o

[If there is no a such that v,

= 0,

then

Pa. r] =

(23.66)

co in (23.66).]

REMARK 23.9. After (23.64.1) and (23.64.2) are calculated for i = 0, the further iterative calculations of (23.64.1) and (23.64.2) may be performed only for those a’s and p’s such that v, # 0 and v p # 0, respectively, . as will be verified by examining the effect of 5, and ipon la and is i

i

i+ 1

i+ 1

After the [,’s and 5,’s converge for those a’s and p’s, we calculate by i

i

(23.64.1) the cis for the remaining a’s. The [,’s for the remaining p’s are put equal to the 5,’s (=O). 0

Solution of the associated maximum-Jow problem: For finding a conductive path, we make use of Labyrinth Algorithm 2 in Remark 20.4 in 420.3.1. We determine a o , B0, . . ., am-l, fim-l and . . ., 6,, as follows :

(i) Let a. be the smallest among the a’s for which v, = 0 and

e,

= r].

237

23. Two-Terminal Characteristics

(If there is no such a, then we should have a solution.) Then put = * . . = a*-, = 0 ,

/ ? o = * * * = p l n -=111, ,

(5,1,+1 = . . . = 6 Z , , l = 0.

(ii) Let i be the smallest number such that u i# O‘and let ,j be the smallest number such that - eUi,,;+ = 0 and J B i + = 0 (.j L 1, p i + j 2 2m). I f v , , ; + ~= 0, then we proceed to (iv), but otherwise, we = I , and then return to the replace p i by p i +.j, set a i + l = v b , and beginning of (ii). I f such a .j cannot be found, then we set P i = nz and ai = 0 and return to the beginning of (ii) (in case i > O), or we proceed to (iii) (in case i = 0). (iii) Set Po = m and l e t j be the smallest number such that [ , o + j = q and v , , + ~= 0 ( j z I , a. +,j 5 m). Then replace a. by a. + j and return to the beginning of (ii). If no such j can be found, we conclude that a conductive path is nonexistent. (iv) Replace p i by p i + j , and set r equal to i. The sequence ( a o , Po, . . ,a,, p,) represents a conductive path. Assigning the unit incremental flow along this path corresponds to setting v,; = pi and vDi = xi for i = 0, . . . , r.

eai

.

This process is repeated until we find an ultimate solution of the problem [in (i)] or a solution of the associated maximum-flow problem [in (iii)].

REMARK 23.10. It should be noted that in the above algorithm, the

pi’s correspond to the +aI)s in Labyrinth Algorithm 2 and the bS’s to part of the IS,’s, whereas we can dispense with the 6,’s because if (and only if) v, # 0, then there is exactly one p such that v,, = a (or v, = p). We can dispense with the signs before the xi’s in Labyrinth Algorithm 2 by virtue of the fact that the direction of the conductivity is evident. The information carried by the 0 ; s is automatically included by the ordering of the p’s. uI)s and

REMARK 23.1 1. In concluding this section, let us add a remark on the computational labor required to solve an H I x nz assignment problem. As careful examination of the algorithm of finding a conductive path (given above) will reveal, no pair (a, fl) is referred to more than once during the labyrinth search, so that we require the labor that is at most proportional to m 2 in order to find a conductive path (cf. Remark 20.5 in $20.3.1). On the other hand, solution of the associated minimum-route problem with regard to a flow configuration with the total number of assignments equal to 5 requires at most + 5) z t3additions and comparisons for the iterative calculation of (23.64.1) [or (23.64.1’)] and (23.64.2), in addition to requiring the labor that is at most proportional to MZ’ for other computations

{(c2

238

V. Linear Network-Flow Problems

(see Reinark 23.9). Therefore, in the worst case where we have to solve rn minimum-route problems, the leading term of the expression of the total amount of labor, i.e., the total number of additions and comparisons, required to solve an H I x I I I assignment problem is nz4/4.

PROJECT SCHEDULING PROBLEM 23.4.4. M INIMAL-COST I n all the examples of application thus far illustrated, " flows" have been given primary physical meanings while tensions have played an auxiliary or secondary role. I n the following. we shall give an example in which tensions arc our primary concern. This problem is a generalization of the one dealt with in 921.5 and it was roughly surveyed in $2 of Chapter 1. We consider a project consisting of a system of jobs represented by a given arrow diagram (cf. $2 and $21.5). As before, a branch of the arrow diagram represents a job, the tension across it represents the negative of the job time of that job, each node represents the event that the jobs corresponding to the negatively incident branches have finished and the jobs corresponding to the positively incident branches are going to start, and the potential at a node represents the time of the corresponding event. We regard the node representing the start of the entire project as the entrance and that representing the completion as the exit, thus making the network a two-terminal network. In addition to this characterization of the network, we shall take into consideration the costs connected with the jobs, i.e., we assume that the cost for a j o b i i to be completed in the time -q, is equal to $"(v],), where $"(q) is a piecewise linear convex function of v] [see Fig. 23.22(a)]. If we Put

5"

4f(~J/dqK

and regard 5" as the flow i n branch b, , then we have the relation between and described by a nondecreasing step-function [see Fig. 23.22(b) and (c)]. Thus we have a two-terminal network of the kind with which we are dealing in this section. The relations among the event times and the job times are exactly like those among the node potentials and the tensions across branches. The difference - q of the potential of the entrance from that of the exit corresponds to the total project time. Therefore, the plan of allocating the times to the jobs i n such a way that the total project cost

<"

=

IC/"(qK)

is as small as possible under the given total project time -q can be obtained by calculating the tension configuration with the given entrance-from-exit potential difference - q. Moreover, by solving the two-terminal character-

239

23. Two-Terminal Characteristics

Job time

Normal duration

'

(a)

) (b)

(C)

Fig. 23.22. Characteristic of a job.

istic problem thoroughly, we can know the relation between the total project time and the total project cost, as well as the lower limit of the project time. Incidentally, if every job (branch) has the $K(qK)whose curve ends with a leftward semi-infinite horizontal segment, we can adopt as the starting flow configuration 5 = 0 (cf. 921.5). 0

In the conventional terminology, the least necessary time for a job is called the crush duration and the time corresponding to the right end of the horizontal segment of the curve $K(qK)[see Fig. 23.22(a)] is called the normal duration. If the value of q is such that the corresponding t is positive, then the jobs (i.e., branches) with nonnull flows constitute a path from the event of start (i.e., entrance) to the event of completion (i.e., exit). Such a path is called a criticulparh, since the alteration of allocation of times to those jobs entails the change of the total project cost.

240

V. Linear Network-Flow Problems

(b)

(a)

Fig. 23.23. Milestones: (a) ts6 td-+e; (b) tr 2 5.

+ e.

REMARK 23.12. Special kinds of events or nodes have their proper names. We sometimes require the schedule of the project to be carried out in such a way that a specified event (or events), say event B, may take place not later than the given time e after the project starts (or, more generally, after another event, say event a, takes place). Such a requirement is realized by connecting a voltage limiter between the nodes nu and nB corresponding to those events; similarly, if we require that event fl should take place not earlier than the given time e after event 01 takes place (see Fig. 23.23). Events of this kind are called milestones. To realize a special case of the preceding conditions, we often make use of the jobs called dummy jobs which have null crash duration and null cost, i.e., which have the characteristic curve shown in Fig. 23.24. A dummy job does nothing but specify the priority between two events and is a mere diode in the terminology in $13.3 of Chapter IV. REMARK 23.13. There arises the question of whether or not we can always construct an arrow diagram representing the partial order relation among the jobs of any given project, and also the question of how to construct it (and, if more than one arrow diagram represents the same partial order, how to construct the simplest among them). The answer to

(b)

(a)

Fig. 23.24. Dummy job.

24 1

23. Two-Terminal Characteristics

the former question is affirmative, as will be shown by giving a method of constructing such a diagram. The latter question seems to concern the skill of the man drawing the arrow diagram, at the present state of the art. Usually, the partial order relation among the jobs of a project is given by a diagram such as Fig. 23.25(a) (or something equivalent), where the job at the head of an arrow must be started after the completion of the job at its tail. We can always convert such a diagram into a network by first writing out the branches corresponding to the jobs separately and then connecting those branches by diodes in such a manner that each diode

S

pletion

(4 Fig. 23.25.

242

V. Linear Network-Flow Problems

Job I

------t

Job 2

(f)

Fig. 23.25. From partial order to arrow diagram.

corresponds to an arrow of the given diagram and connects the branches in the same direction as that arrow connects the corresponding jobs. The network constructed from the diagram of Fig. 23.25(a) is shown in Fig. 23.25(b). This is already one of the arrow diagrams, or the networks, representing the given project, as is easily seen. However, in most cases, a network constructed in this way is awkward with many unessential dummy jobs (or diodes). Therefore, we try to simplify it as far as possible by the equivalence transformations of networks expounded in $19 of Chapter IV. The first step is to remove the unessential diodes which are placed in series with the branch representing a job [transformation of type 2.3 in $19; see also Fig. 23.25(c)], and then we look for the possibility of applying the transformations of type 2.3, especially the star-mesh transformation. In the above example, we can apply the mesh-to-star transformation [the case when r = 4 illustrated in Fig. 23.25(d)] to the part enclosed by a dotted line in the network of Fig. 23.25(b). The resulting network is shown in Fig. 23.25(e), to which we apply the transformation of removing the

24. General Linear Network-Flow Problems

243

series diodes (as mentioned above) to obtain the network of Fig. 23.25(f). The network of Fig. 23.25(f) contains only one dummy job and is the simplest arrow diagram for the given project. 24. GENERAL LINEAR NETWORK-FLOW PROBLEMS

In this section we shall take up the methods of solving the general form of the network-flow problem as defined in Chapter IV, i.e., the problem of finding a compatible pair (i.e., a pair satisfying all the branch characteristics) of flow and tension configurations in a given network whose branches have monotone nondecreasing step-function characteristic curves. 24.1. Solution Algorithms

We shall explain two kinds of algorithms, one of which solves a problem by using the two-terminal characteristic algorithm in $23.3 only once, and the other which modifies an initial noncompatible pair of flow and tension configurations, step by step, until a compatible pair is finally arrived at. We shall call the former the direct method and the latter the step-by-step method. Which method is more efficient will depend on circumstance. 24.1.1. DIRECTMETHOD We may assume that the network N under consideration consists of a finite number of diodes, current sources, and voltage sources, since any linear network-flow problem can be expressed in such a form (see $13.3 of Chapter IV). In that network, we split each current source into two current sources of the same value connected in series [transformation of type 2.1(3) in $19 of Chapter IV; see also Fig. 19.41, gather the middle points of those split current sources into a single point A [transformation of type 2.1(4)], and then divide the point A into two points A' and A" in such a way that every current source incident to A' makes a flow from A' to a node in the network, whereas every current source incident to A" makes a flow from a node in the network to A". Moreover, we replace each current source by the series connection of a diode and a current limiter of the same value as that current source and regard the network N' thus obtained as a two-terminal network with A' as the entrance and A" as the exit. (This process of transformation is illustrated in Fig. 24.1.) It will need no rigorous discussion to see that if and only if N has a solution, then the maximum value of the total entrance-to-exit flow 5 through N ' is

244

V. Linear Network-Flow Problems

equal to the sum of the values of the current limiters incident to A' (or to A"). Furthermore, if N has a solution, then the flows and tensions in the branches of N coincide with those in the corresponding branches of N', except in the current sources of N and the current limiters (and their series diodes) of A'''.

N

Nt

Fig. 24.1. Direct method.

As the starting flow configuration for the two-terminal problem of N', the null configuration 6 = 0 may be adopted for the following reason. If the 0

associated minimum-route problem yields a solution q that is compatible with 6 ( = O ) , there is no problem. On the other hand, the nonexistence of 0

245

24. General Linear Network-Flow Problems

solution for the associated minimum-route problem implies the failure of the nonnegative loop-span condition in regard to the flow configuration 6 = 0. Moreover, a closed path in N' around which the sum of the spans 0

of the branches is negative should consist of diodes (placed in the positive direction) and voltage sources which were originally in N.However, the span of a diode or a voltage source in the positive direction is independent of the flow in it so that we cannot find a tension configuration in N (or in N') compatible with any flow configuration. Therefore, in this case, we can conclude that a solution in the given network N does not exist. 24.1.2. STEP-BY-STEP METHOD As we did in $16.3 of Chapter IV, we shall denote by C,'(C, q ) the monotone nondecreasing step-function which passes the point (t,q ) and is as near the characteristic curve C, of branch b, as is possible (cf. Fig. 24.2). 9.1

*--------

(5, rl) I

lC.'(t> 'I)

-

I

I I I I I

I

II

c,

-

5"

I I I

Fig. 24.2. C. and C/o, 7).

Let the given network be

N = N ( 0 ;c,,... , C,),

(24.1)

where 8 is the graph representing the topological structure of the network and C, (K = 1, . . ., n) is the characteristic curve of branch b,. In the following, we shall make use of the two-terminal networks such as N ( 0 ;C,, . . ., CK-,, ref., C r + , , .. . , C,) with a certain branch b, of N as the reference branch.

246

V. Linear Network-Flow Problems

We may start the solution algorithm from any pair of flow and tension configurations (5, q) so long as they satisfy the continuity conditions 0

0

(Kh.F.) and (Kh.T.). We first calculate the two-terminal characteristic curve of Jv-,

= "6;

ref., C2'(t2, 1 2 ) , 0

0

* * *

,C,1(5", 1,>1, 0

(24.2)

0

with the (5,q) as the initial configurations [which obviously satisfy the 0

0

CK'(tK, qK)'s]. The two-terminal characteristic curve [which is the set of 0

0

points on the (t', - q l ) plane] is, when drawn on the(t', ql) plane, amonotone nonincreasing step function, which we shall denote by C,, calling it the iriverted tlt*o-terminalCharacteristic of N l . If El intersects C,, then we choose a pair (5,q) corresponding to a point ({I, q l ) ~ C n l C,. 1

I

1

1

[In passing, we need not determine the entire curve C,, but we may stop the calculation as soon as an intersection of and Cl- is found since our aim is to obtain a pair (5, q) such as the above.]

el

1

1

Obviously, (5, q) is a solution for 1

1

.Ir[@; c,,c2'(t2? 0

c.'(5",

q2)7 0

0

?.)I*

(24.3)

0

Then, with (6, q) as the initial pair, we solve the two-terminal characteristic 1

1

problem for

to obtain the inverted two-terminal characteristic curve C2 of choose (5, q) such that (t2,q 2 ) E C2 n C 2 . 2

2

2

N2, and

2

(5, q) is a solution for 2 2

In general, the inverted two-terminal characteristic curve CK+lfor A+1 = "Q;

c,,.

*

f

9

c, , ref., C:+2(tK+2,q2+2), c,ltr.,rJl * * *

K

9

K

K

K

(24.6)

is calculated with

(5, q) as the initial pair, K

K

and (

6,

K+1

q ) is chosen such K + l

247

24. General Linear Network-Flow Problems

that (

tK+',qK+')

K + l

E C ~ n+Cr+l ~ (see Fig. 24.3). Then, ( 6 , q ) is a

K+1

K + l

K + l

solution for

"6; C1, - * * CK CK+1, C:+2(tK+ 9

~,+z),***

3 '

K

Cn'(Y, 1 n ) I

K

K

(24.7)

K

and can be adopted as the initial pair when solving the two-terminal characteristic problem for

(24.8)

---_1 I

I-----

-I

m .

T(I 5"+'. K+l

%+I) I+l

'5""

I

L--, I I

1 L----

Y

cK+,:Inverted two-terminal characteristic of

Fig. 24.3. CKand

Proceeding in this way, if calculate the C, for

N,+I

eK.

E, intersects C,

for

IC

N,= N ( 6 ;C1,. .., Cn-l, ref.)

.

= 1, . ., n

- 1, we (24.9)

6 , q ) as the initial pair. Finally, if enintersects C,, we choose a pair (g, fi) = (6,q) such that (P,1,) E C, n C,. The pair of flow and n n n n tension configurations (f, fi) is obviously a solution for the original with (

n-1 n-1

network JV of (24.1). Thus, by solving the two-terminal characteristicproblem as many times as the number of branches according to the preceding algorithm, we can attain a solution so long as every E, intersects C, . We can show a fortiori that if E, does not intersect C, for a certain K, the nonexistence of the solution is asserted.

248

V. Linear Network-Flow Problems

e,

Let us consider the case where n C, = 0. By virtue of the monotonicity of C, and C,, there are two possibilities, in one of which we have J , n J, = 0 and in the other of which we have I" n I" = 0, where I" (or P) and J, (or J,) are the projections to the 5 and the q axes of C, (or C,), respectively (see Fig. 24.4). Since Jlr, can have no feasible tension configuration if the negative of the tension across the reference branch b, is outside J , , there is no feasible tension configuration in JlrC(r,;C,,...,CK --l)CK)C:+1(

~ , + 1 ) , . . * , C n lr", ( qn>I (24.10)

tK+'7 K - 1

K-1

K-1

K-1

in the former case. Furthermore, for any (t,q), the projection .I,(< q) of , C,'(<, v ] ) to the q axis includes the projection Jn of C, , as follows from the definition of C,'(t, v ] ) . Therefore, if the network of (24.10) has no feasible tension configuration, then the network of (24.1) can have no feasible ten-

(c)

(d)

Fig. 24.4. Cases where there is no solution.

249

24. General Linear Network-Flow Problems

sion configuration, and consequently, no solution; similarly for the case where f" n I" = 0.

REMARK24.1. In most practical cases, the preceding algorithm is much simplified as follows. If, for a certain K , it happens that ( t", q,) E C,, K-1

then we may go directly to JV,+~with (5, q) = (

K-1

5 , q ) without solving K-1 and A (2K + 2), it happens that

K

K

K-1

the problem for N,. If, for a certain K (tA,q,) E C A ,then we may replace the CA"sin JV,+~, .. ., NA.-l by CAand K

K

omit solving the problem for N AFurthermore, . when, in the course of , the point (tA,qA) [which travels on C,'( tA,qA) as q,) calculating

(r,

eK

K-1

e,]

K-1

gradually draws the curve arrived on C, , then we may at once replace the C,' in N Kby CAand use the latter curve in the subsequent calculation. By so doing, we can omit solving the problem for N A.

REMARK 24.2. This step-by-step method is essentially the same as the out-of-kilter method of Ford and Fulkerson [l]. 24.2. Applications

We shall take up a few arbitrary examples of application which should not be considered as typical applications. In fact, the techniques developed in this section cover too wide a region (the whole region connected in some sense with a network-flow problem) to systematically treat here. 24.2.1. APPLICATION TO NONLINEAR PROBLEMS A nonlinear network-flow problem (Le., the problem where some C,'s are not step-functions but general nondecreasing curves) can be solved by successive approximation using iteratively the algorithm for a linear problem. The following treatment is in close connection with $18 of Chapter IV. Let us suppose that a network N o = N ( 6 ;C1, ..., C,) is given. Since the existence or nonexistence of solution is easily checked by means of the projections I" and J, of the characteristic curves C, to the t and the v] axes (cf. $15.4 and $16.4 of Chapter IV and $20.3 and $20.4 of Chapter V), we may assume that we have a (noncompatible) pair consisting of a feasible flow configuration and a feasible tension configuration (5, q), denoting by

(s, 4) a solution (which is not yet known but whose existence is assured). 0

0

0

0

As we did in Theorem 18.2 (in $18 of Chapter IV), we denote the horizontal and the vertical distance from (tK, q,) to C,, respectively, by 6" and E, 0

0

250

V. Linear Network-Flow Problems

(see Fig. 18.4), where all the SK's as well as the E,'S are finite by virtue of the feasibility of 6 and q. Then, by Theorem 18.2, we may assume that 0

0

15" - ["I 5 6 0

0

lq, - +,I

and

0

0

5E

0

for every K ,

(24.11)

0

where 6=cSK O

i.e., the point

(tK,4), 0

R,

=

0

&=I&,,

and

K

O

(24.12)

K

lies in the rectangle

0

( ( 5 . q ) I 15 - 5"l 5 6, I? - rlKl 5 E ) 0 0 0 0

n (IK x J , ) .

(24.13)

Next, let C, be a monotone nondecreasing step-function such that the I

horizontal distance from a point on C, n R, to C, is not greater than 6" 1

0

and the vertical distance is not greater than

1

E,

and that (tK, q,) is nearer to 0

1

0

C, than to C, (outside R , , C, is arbitrary so long as it is in I" x .IK), 1

0

where the 6"'s and c,s' 1

1

are positive numbers satisfying the conditions

1

S=CS"
K

and

00

1

with a certain positive constant

Y

E = ~ E , < ~ E , 1 K 1 00

(24.14)

less than unity. Applying the algorithm

0

for the linear problem, we can obtain a solution

(5, q) for 1

1

J1 E N ( 6 ; cl, . . . , C"). 1

(cK,qK) lies in R ,

The point

1

1

1

by virtue of Theorem 18.2, and, again by

0

Theorem 18.2, it satisfies

15" - <"I 5 8 1

1

and

1

Irl, - riKl 5 E , 1

1

(24.15)

1

where (& fi) is a solution for .No.[Strictly speaking, it is possible to find a 1

solution

1

(6, q) for N l such that (tK,q K ) lies 1

in R, for every K and also

1

a solutionl(i$lfi) for N o satisfying (24.15). This 1

n

(6, q). The 0 0

be

(c, fi) mentioned in the above in connection 1

1

different from the solution with

0

(t,Q) may or may not 1

n

method of modifying a solution

(5, q) for 1

1

Nl, where

251

24. General Linear Network-Flow Problems

some

(F,q,) 1

is not in R,, into a desired one is given in the proof of

1

0

Theorem 18.2.1 The subsequent procedure is similar to the foregoing. We define R , = {(t,?)I It - Yl 5 6, 1

1

IYI - ?,1I

1

s E l n (1" x J , )

(24.16)

1

and construct the step-functions C, such that the distance from a point on 2

C, n R, to C, is not greater than 6" horizontally or 2

(t",q,) 1

2

1

E, 2

vertically and that

is nearer to C, than to C,, where the 6,'s and

1

2

2

are positive

E,'S 2

numbers satisfying the conditions (24.17) with a certain positive constant r less than unity. Then 1

(p,9,) E R, . We 1

1

1

solve the linear network-flow problem for the network N

2

= N ( 8 ;c,,.. . ,C") 2

(24.18)

2

to obtain a solution (6, q) such that (t",q,) E R, and that 2

56

2

2

2

2

2

1

lq, - 9,l

and

2

2

2

5E

(24.19)

2

with a solution (f, fi) for N o .Proceeding in this way, we have a sequence 2

2

of rectangles (R,, R,,. . .) for every 0

K

and a sequence of pairs

1

[(t,q), 0 0

(6, q), . . .] as well as a sequence [(t,fi), (5, fi), . . .] such that (5, q) is a 1 1 0 0 1 1 i i solution for Niand (f, fi) for N o and that i

i

(24.20) and (24.21) The length of the horizontal side of R, is not greater than two times i

6 where i

252

V. Linear Network-Flow Problems

and, similarly, the length of the vertical side is not greater than two times E where i

E i

If the series 1 + r

<

(ij' r)c. j=O j

(24.23)

0

+ r r + r r r + - . . converges (e.g., if

all r

0 1 2

0 1

0

=

3),

then,

i

due to (24.20), the sequence of rectangles ( R , , R, , ...)converges to a point 0

1

fi,), and, therefore, due to (24.21), [(p, fi,), (p,fi,), . ..] converges 0 0 1 1 to the same point. Since (&Q) is a solution for N o and since the set of ([,,

i

i

solutions for M o is closed, (4,i)) is also a solution for N o . Thus, we can construct a solution 4) of a nonlinear problem by successive approximation with the help of the algorithm for the linear problems.

(s,

24.2.2. SENSITIVITY ANALYSIS

Suppose that we have a solution as well as the value of the objective function Q, or Y for a network N ( 6 ;C1, ..., C,J, and we want to investigate the behavior of the solution as well as the value of the objective function as the characteristic C,'s of the branches are gradually altered. This kind of study in the dependence of the solution on the branch characteristics is sometimes called the sensitiuity undysis. From this standpoint, the descriptions in $16.3 of Chapter IV are regarded as the fundamentals of the sensitivity analysis, and the algorithm developed in $24.1 of this chapter is the basic technique of the sensitivity analysis for the linear network-flow problems. 24.2.3. ASSIGNMENT PROBLEM

Let us return to the assignment problem treated in 923.4.3. This problem may be described as one of obtaining a set of values of the variables C U P (a = 1, . . . , m ;fl = m 1, . . . , 2m) which minimizes

+

(24.24)

under the conditions

YB=O or 1,

cp=1 c r"p= 1

t==

Zm

(u=l,...,m),

b=m+l

m

EZ

a= 1

(24.25)

( p = rn + 1, ..., 2m).

24. General Linear Network-Flow Problems

253

A network representation of this problem was given in $23.4.2 (see Fig. 23.7), but another representation shown in Fig. 24.5 will be adopted here. consists of 2m The network of Fig. 24.5 (which will be denoted by Nm) nodes nl,. . . , nm, nm+lr.. ., n2m and m2 + in branches. These branches are classified into two groups, one consisting of m2 branches and the other of in branches. A branch of the first group connects na to ns and is the series connection of a voltage source of value eas (a= 1, .. . , m ; = in + 1, . . .,2m),a diode, and a current limiter of unit value, while a branch of the second group is a current source of unit value connecting nm+a to na (M = 1, . . . , m). The flow in the branch connecting n, to ns is taken as cp.[The first condition of (24.25) automatically follows from the solution algorithm; cf. Remark 23.3.1 We designate as A’”, the network obtained from N,,, by replacing the current sources incident to n,,,, . .., nm by the current sources of value zero, i.e., by open-circuiting those current sources. We have the obvious solution for N o , i.e., (6, q) = (0,O). Therefore, we can modify step by 0

0

1

Fig. 24.5. Assignment problem.

254

V. Linear Network-Flow Problems

step a solution

(6, q) a

for .Nu to (

so~ution(6, q) = (6, tj) for N,. 111

6,

a i l

a

Ill

The procedure for determining (

6,

h+l

finally getting a

q ) for a+l

q ) from

(6, q) is as follows. We a

h+l

a

regard the ( 2 + I)st current source (i.e., the one connecting n m + a + l to na+& of .Ma or (its value being 0 in N , and 1 in N z + l as ) if it were the reference branch, where <"I = tm'a+lis the flow in the reference branch. To represent the flow configurations, we shall use the vector v = (v,, . . .,vZm) that we used in $23.4.3. We may adopt the flow configuration 6 as the initial configuration and solve the associated minimum-route (I

problem for that two-terminal network first. To do this, we set

if

Cs=co

p#rn+E+l;

(24.26)

0

=O;

[ , = co

0

f o r a l l c l( = l,

..., m),

0

iteratively by means of (23.64.1) and (23.64.2)

and then calculate (, and i

i

until we have [, = r

5, = pa

Cs = Cs = [,

and

r

r+l

The tension configuration q is calculated from the a+

p.

for all CL and

(24.27)

r+l

ta's

and

lp's thus

1

determined, where the tension across the reference branch is (24.28) Then we can find a conductive path from n,+, to nm+a+l along which we can assign the unit incremental flow to obtain 6 . The algorithm for a+ 1

finding a conductive path is essentially the same as that given in 823.4.3 where we put u0 = 2 + 1 and search for a path such that fir = rn 5 1. As will be seen without difficulty, in so doing we may exclude the nodes n a + 2 , ..., n ,, na+,+z, ..., nZm, as well as all the branches incident to those nodes, from our consideration, i.e., we may do the calculations of [ a a n d ( a o n l y i n r e g a r d t o a = 1,..., E + l a n d p = r n + 1,..., r n + E + l

+ +

i

i

and search for a path with aiand and rn 1, . . ., rn + 5 1.

+

+

pi chosen only from among 1, .. . , 5 + 1

REMARK 24.3. It should be noted here that although we made the correspondence between na and nm+a in the above, connecting the pair of

255

24. General Linear Network-Flow Problems

nodes under that correspondence by a current source, we may in general make use of any one-to-one correspondence between the set of n,‘s and that of n i s . The preceding algorithm can then be regarded as an extension of that in 523.4.3. In fact, we shall have an algorithm equivalent to the former if we replace (23.63) by (24.29): [a 0

=0

for an arbitrary p such that va

= co

for all the other p’s;

[, = co

= 0,

(24.29)

for all a,

0

and (23.66) by (24.30) r] = arbitrary

one of the [,‘s for which v,

= 0,

(24.30)

leaving (23.64.l), (23.64.2), and (23.65) unchanged. However, the value of r] is not necessarily increasing as the solution algorithm proceeds, and it can sometimes even be negative (in which case the value of decreases). REMARK24.4. The technique of the kind we have developed here is applicable to all the multientrance-multiexit problems. The reader should investigate the case of the Hitchcock transportation problems (cf. Kurata [l]).

Appendices

APPENDIX 1 : FARKAS’ THEOREM

The matrices and the vectors we are going to deal with are those consisting of real elements. It is a well-known fundamental theorem in linear algebra that: For a given constant matrix uii (i = 1,. . ., m ; j = 1, .. ., n), a given constant vector p j ( j = 1, ..., n), and a variable vector xi ( j = 1, . .., n), if

c n

i= 1

UijXj

=0

(i = 1, . . , , rn)

(Al.l)

implies n

CPjxj=O,

(A1.2)

j= 1

then there is a constant vector yi (i = 1, . ..,m) such that rn

pj =

cyiaij

( j = 1, . ..,n ) .

(A1.3)

i=l

Conversely, if (A1.3) holds, then ( A l . l ) implies (A1.2). The extension of this theorem to the case of inequality relations, 256

257

Farkas’ Theorem

established by J. Farkas [l], plays the fundamental role in the theory of mathematical programming. Theorem ofFurkus. Consider a matrix aij (i = 1, . . . , m; j = 1, . ..,n) and a vectorpj ( j= 1, .. .,n). If every vector xi ( j = 1, ...,n) that satisfies the inequalities n

1a i j x j >= o

for all i ( = I ,

..., rn)

(A1.4)

j= 1

also satisfies the inequality (A1.5)

then there exists a vector with nonnegative components zi( 20; i = 1, ..., m)suchthat

c rn

pj =

( j = 1,

ZiUij

. ..,n).

(Al.6)

i= 1

The converse is also true. Before beginning the proof, we note that the aforementioned theorem in linear algebra follows from the theorem of Farkas. In fact, if we define a matrix b, (i = 1, . . ., 2 m ; j = 1, . . . , n) by

bIJ. . = aIJ. .

(i = 1, ..., m ;

bI.J . = ai-rn.j

(i=m+l,

j = l , ..., n ) ,

..., 2 m ;

(Al.7)

j=1, ..., n),

then ( A l . l ) is equivalent to

c b i j x j2 0 n

(i = 1, ... , 2 m )

(A1.8)

j= 1

so that (A1.8) implies (A1.5). By Farkas’ theorem, there exists a vector with nonnegative components zi (i = 1, . . ., 2m) such that 2rn

pj=Xzibij i= 1

If we set yi = zi

- zi+,,,

(j=I,

..., n).

(i = 1, .. . , m),

we have from (A1.9) rn

pj =

which is exactly (A1.3).

1y i a i j

1= 1

.

( j = 1, . . , n),

(A1.9) (A1.lO)

258

Appendix 1

Although Farkas’ original proof is constructive and elementary, it is of considerable interest in that one can find there a technique similar to the simplex method (which was lately developed in connection with linear programming). Let us therefore describe first Farkas’ original proof with some notational modifications. PROOF1. We shall give a proof to the first part of the theorem since the “converse” part is self-evident. We resort to the mathematical induction with respect to n. It is easy to prove the case where n = 1. For an 11 2 2, let us suppose that we are given a matrix a j jand a vectorpi satisfying the assumption of the theorem, and prove the existence of zi with the desired property. Since, if pi = 0 for all j , (A 1.6) holds with the null vector zi = 0 (i = I, . . . , m), we assume without loss in generality that at least one p i , say pl, is nonnull. Then we change the variables (xl, x2,. . . ,x,) to (5,x2, ..., xn) by (A1.11)

(Al.ll) and rewrite (A1.4) and (A1.5) in terms of the new variables. (A1.5) is rewritten simply as 5 >= 0. (A1.12) I f we denote by I + , Io, and I - the sets of indices such that plail > 0, p l a i , = a,, = 0, and pla,, < 0, respectively, and if we define a new matrix b j j ( i = I , . . ., nz;.j = 1, .. ., n ) by bij

= (aij/ai1)Pi - P j

( i E I + ; j = I,

- aii

(iE1’;

=

-(ajj/uj,)pl +pi

(iE

..., n),

..., n), = 1, . . . , n),

j = 1,

I-; j

(A1.13)

then (A1.4) is rewritten as

n

j=2

-5

+

b i j x j2 0

( i E I0),

b i j x j2 0

(i E

(A1.14)

Z-),

j=2

where it is noted that b i , = 0 for all i according to (A1.13). Hence, (A1.14) should imply (A I. 12). Here we observe that I + is not empty, for if I + were

259

Farkas' Theorem

empty, there would be a negative 5 which satisfied (Al.l4), together with certain x2, ..., x,, say with x2 = -.-= x,, = 0. Next, we introduce another matrix cij [ i E I + u Zo u ( I + x Z-); j = 2 , . . ., n] with m+ + m0 m + m - rows and n - 1 columns as follows, where m+,mo, or m- is the number of elements of I + , Zo, or I - :

+

(i

cij = bij

€

I + u zo; j

[hEI+;

C(h,i),j=bhj+bij

= 2,

. . .,n ) ,

(h,i)EI+

i€z-;

XI-;

j=2

,..., n]. (A 1.15)

Then, the conditions

g+

2c i j x j 2 0

2c i j x j 2 0

(i E I+),

j=2

( i E Zo

uI+ x I-)

(A1.16)

j=2

must imply (A1.12), for if (A1.16) were satisfied by a set of values ( f , X,, . . . ,Xn) with 5 c 0, then, setting

f = -min C c i j ~ j -=c c i o j X j 6 [ ( < O ) , isl+ j=2

(A1.17)

j=2

we should have

=

-f

+

n

b i j X j2 0

for all i E I - ,

(A1.18)

j=2

which, however, would mean that ( f , X 2 , . . .,X,,) satisfied (A1.14) and consequently, that f 2 0 [by (Al.l2)], in contradiction to (A1.17). We then consider the system of inequalities:

c c i j x j2 0 n

[i

EI

+ u

I0

u (I+ x I-)].

(A1.19)

j=2

There exists an i E I + such that every (x2, . . . , x,) satisfying (A1.19)

c c i j x j ; for if there were a (X, , .. .,X,,)corresponding to each n

nullifies

j=2

i

i

i E I + such that it satisfied (A1.19) and also CijFj

j=2

i

> 0,

(A1.20)

260

Appendix 1

then

x i = i s1 Z~ l +i

..., n)

(j=2,

(A1.21)

would satisfy (A1.19) with all the inequality signs replaced by the strict inequalities for i E I + , which would mean that (5, Xz , . . . ,Z,) with

t = - min

c c i j x j c 0,

(A1.22)

isI+ j=2

satisfied ( A l . 16) to yield the contradiction. Thus, it has been shown that (A1.19) implies (A1.23)

By the induction hypothesis, there is a vector with nonegative components upi [ 2 0 ; i E I + u Zo u ( I + x 1-)] such that -ctj =

Ci w i c i j

( j= 2,

. . . , n).

(A1.24)

Rewriting (A1.24) in terms of b,,, we have m

- btj = C ui b i j ,

(A1.25)

i= 1

where the 1);s are nonnegative numbers defined by ui = wi = wi

+h c W(i,h) sl-

z+),

(i

E

(i

E lo),

(i

EI-).

(A1.26)

Rewriting further in terms of a i j and p i , we have

For j

= 1,

the same form of the relation as (Al.27) identically holds re-

26 1

Farkas’ Theorem

gardless of the values of ui . Therefore, rearranging relations (A1.27) and that of j = 1, we have

(A1.28) where UIJ

=1

+ c + v i- c isI

ui

=1

+

+ 1 wi > 0, isI+

ieZ-

> 0,

U1

=(Pl/atl)(l

ui

= (Pl/ail)ui 2 0

(iEI+,

=uiqo

(i

=

Vi)

E IO),

(i E z-),

20

-(pl/Ui&i

(A1.29)

i # t),

or we have m

pj=

.c

zi a i j

1

a=

( j= 1,.

.., n),

where Zi

= Ui/Uo

20

(i = 1, . . ., m).

Q.E.D.

(A1.30)

In contrast with Farkas’ original proof, a nonconstructive proof from the modern standpoint will be given in the following. (Farkas’ theorem can be proved also as a by-product of the simplex method in linear programming, which, however, we shall not mention here.) PROOF2. We shall prove the contraposition of the (first half of the) theorem, i.e., we shall prove that, if pi cannot be represented in the form (A1.6), then we can find a set of values (21,. . . ,a,,) such that n

1a i j R j 2 _ 0

forall i ( = l ,

n

..., m)

j= 1

j= 1

Let us consider the closed convex set space (cf. Chapter 11) defined by m

(X =

(A1.31)

and (X

in the n-dimensional Euclidean

I

((51, . . . , en) I c j = 1 zi aij for some nonnegative zi)s . i= 1

(A1.32)

As usual, we define the distance IIp - 611 between two vectors p = (pi) and 5 = (tj) by (A1.33)

262 Then I/p -

Appendix 1

is a continuous function in the tJ’s,and, by assumption,

l l ~ - 6 1 1 ~ > 0 if ~ E C .

(A1.34)

Furthermore, if

(Al.35) j= 1

then we have, by the ordinary triangular inequality, IIP - 511 =

11511 - llPll > IlPll = IIP - 011,

(Al.36)

where

..., 0)EC.

O=(O,

Hence, the minimum value of IIp - 511’ (with regard to 6) in C, if such a one exists, is not outside the region (5.’ which is the intersection of (5. and the region (5 I 11511 6 2IIpII). Since C’is obviously a bounded closed convex set, i.e., it is compact, the minimum is actually attained at a point E 6‘. Then we shall show that the vector S = (21, . . . ,a,) defined by S=t-p

or

( j = 1,

gj=tj-pj

has the desired property. Let us consider, for an arbitrary i, the vector

<,(A)

=

..., n)

&(A) defined by

t j + La,..

Since &(,belongs I) to C for every nonnegative

(A1.37)

(A I .38)

A and since

n

+ 22 c a i j g j+ A2

IIP - 5(~>11’ = 118112

j= 1

a;

(Al.39)

j= 1

takes its minimum value at t(0)= 6, we must have n

( d / d A ) Ilp - 6(A)lli=0= 2

j= 1

aij)Zj2 0

or

c n

Nij)Zj

2 0.

(Al.40)

j= 1

Next, let us consider the vector &,I) defined by

Ej(A) = A t j = t j + (A - l)tj

(A1.41)

Equisignum Decomposition of a Vector

263

Since [(A) belongs to 6 for every nonnegative I and since

takes its minimum at [(l) =

t, we must have j= 1

or n

1 t j A j = 6. P = 0.

(Al.43)

j= 1

On the other hand, we have from (A1.34), (A1.37), and (A1.43) IIp - Ell2

=

-@ -

t)

*

P = - p .P > 0

or n

C p j A j < 0.

(A1.44)

j= 1

(A1.40) and (A1.44) show that P = (Al, ..., A,) property

has the desired Q.E.D.

APPENDIX 2: EQUISIGNUM DECOMPOSITION OF A VECTOR

Let us consider a vector space, or module, V(R, d), with 23 = {bl,. . . , b,} as the basis and the field R of real numbers or rational numbers as the domain of coefficients (cf. $7 of Chapter 111). Let W be a linear subspace, or submodule, of V(R, B), i.e., a subset of V(R, 23) with the property (A2.1) : if G a n d g ~ W and a , a ~ R ,then a ( + a t ~W. 1

2

As in Chapter 11, we define car+ 6,car-

(A2.1)

22

11

1 2

5, or car 6,where

as the subset of 23 such that 5" > 0, t" < 0, or t" # 0, respectively. Let us further introduce the symbol 3 to denote the relation between a pair of vectors in V(R, 23) defined by (A2.2):

6 36 1

2

if and only if car'

6 2 car' 6 1

2

and car-

6 2 car- 6. 1

2

(A2.2)

264

Appendix 2

The relation 3 will be called the equisignum inclusion, and has the following property :

6 9 6 for any vector 6 in V(R, b), if 6 3 6 and 6 3 6, then 6 3 6. 1

2

2

1

3

(A2.3)

3

A nonnull vector 6 [E V(R, b)] is said to be minimal in W if 6 E W and if no nonnull vector E,' such that car 6 2 car E,' belongs to W. Obviously, if 6 is minimal in W, so is a6 for any nonnull a in R. The following theorem, which is closely connected with the basis theorem in linear programming, was also suggested by Farkas [l]. Theorem of Equisignum Decomposition. Every nonnull vector 6 in W can be expressed as

6 = E, +

.**

1

where the 4's are minimal in W and

+ E,

(A2.4)

I

6 3 6 for every i.

i

i

To prove the theorem, we first establish the lemma. Lemma. For any 6 in W, there is a 5 such that 6 3 5 and that 4 is minimal in W. PROOFOF THE LEMMA. If 6 itself is minimal in W, then we may put 4 = 6. If it is not minimal in W, there is a nonnull vector E, E W such that 1

car 6 3 car 6,

(A2.5)

1

6 n car' 6 or car- 6 n car- 6 is 1 1 nonempty (because otherwise, we may adopt -6 instead of 5). In this case

where we may assume that either car'

1

1

we can find a maximum positive a (E R) such that 1

(A2.6) and for which we have car 6 z) car 5.

(A2.7)

1

5 belongs to W, and it is not null by virtue of (A2.5). If 51 is minimal in W, 1

265

Equisignum Decomposition of a Vector

then we may put nonnull

5 = 5. If 5 is not minimal in W, we can find similarly a

5 such that

1

1

2

(A2.8)

Proceeding in this way, we have a sequence of nonnull vectors

(6,g, g, ...) in W such that 1

2

6 3g 3g 1

car 6 3 c a r 5 3 c a r < . . .

and

*..

1

2

.

(A2.9)

2

However, the sequence cannot continue infinitely since 23 is finite. Therefore, we have a minimal for a certain I, which we may adopt I

as g. Q.E.D. PROOF OF THE THEOREM. If the given 6 is minimal in W, we may put I = 1 and 6 = 6 to obtain the desired decomposition. If 6 is not minimal 1

in W, the lemma assures that there is a minimal 5 such that we determine the greatest positive number

CI

6 3 6. Then 1

1

such that

1

6 3 6 - CI

6 = g.1

(A2.10)

1 1

5 defined by (A2.10) is then nonnull (if it were null, 6 would have been 1

minimal in W) and satisfies car 6 3 car 5.

(A2.11)

1

If 5 is minimal in W, then we may set 6 = 5 and I = 2 to obtain the desired 1

2

1

decomposition :

6 = c16 + 6. 1 1

2

If 5 is not minimal, we can find a minimal 5 such that g-3 t.and determine 1

2

the greatest positive number

CI

1

2

such that

2

g3g-at3g 1

1

2 2

(A2.12)

2

where 5 is nonnull and satisfies 2

car g 3 car 5. 1

2

(A2.13)

266

Appendix 2

Proceeding in this way, we have a sequence of minimal vectors

(5,5, ...) 1

as well as a sequence of nonnull vectors

0

-

g=<-ag i

(5 = 5, 5, 5, . . .) such that

i-1

i

2

(A2.14)

(a>O),

i i

F35, i-1

1

2

i

(A2.15)

535, i-1

i

and car 5 2 car 5. i- 1

(A2.16)

I

The sequences, however, cannot continue infinitely because of (A2.16) so that for a certain I, 5 will be minimal. We then set 5 = 5 . From (A2.14), I- 1

I

1-1

we have

and, from (A2.15) and (A2.3),

thus obtaining the desired decomposition by rewriting

CI

6 as 6.

i i

Q.E.D.

i

REMARK A2.1. The equisignum decomposition of a vector is not generally unique. REMARK A2.2. If we adopt the ring of integers as the coefficient domain instead of the field of reals or rationals and if we leave the definitions of the relevant concepts as they are in the foregoing [except for the definition of a submodule M hich should be strengthened by requiring the condition if ~6 E W (M # 0), then E,E W, (A2.1‘) in addition to (A2.1)], then we have the theorem slightly modified as fol I0w s.

5 in W can be expressed as a5 = 5 + + 5,

Theorem. Every nonnull vector

*.*

0

1

I

where a is a positive integer, the 5’s are minimal in W, and i

0

53 5 I

for all i.

(A2.17)

267

Theory of Linear Resistor Networks

The proof will be carried out by extending the coefficient domain to the quotient field (Le., the field of rationals), performing the decomposition in regard to the extended domain, and then clearing the common denominator. Moreover, if every minimal vector is an integral multiple of a simple vector (i.e., a vector each of whose components is 0, 1, or - l), then the theorem holds with no modification. This is verified by recalling that the foregoing proofs (of the lemma as well as of the theorem) are formally valid also in this case since the a’s are determined to be integers. i

APPENDIX 3: THEORY OF LINEAR RESISTOR NETWORKS

Using the terminology of $13.3. of Chapter 111, we shall call a network JI’ consisting of current sources, voltage sources, and linear resistors a linear resistor network. In spite of the adjective “linear,” the convex functions (cp and $) connected with a linear resistor are quadratic functions so that the network problem of this kind belongs to “ quadratic programming” rather than “ linear programming” from the standpoint of mathematical programming. In the following, we shall denote by B,, 23, , or 8,,respectively, the set of current sources, the set of voltage sources, or the set of linear resistors (so that we have 23 = BCu bvu 23,, where 23, n 23, = 23, n 23, = 8,n 23, = @), and by n,, n,, nR, or n the number of elements of dC,dV,23,, or 23 (so that we have n = n, nv n,). Furthermore, we assume that 23, is cutset-free and 23, is loop-free, since otherwise, no feasible flow or voltage configuration will be found if we assign arbitrary values to the current and voltage sources. Then, we can choose a treeon the graph 8 of the given network JI’ such that cotree pair (2,T) 2 2 23, and T 2 BC(see Theorem 9.8 in $9 of Chapter HI), where it is assumed that among the m branches of 2,the first m, = m - n, branches belong to 23, and the remaining n, constitute dV,whereas among the k = n - m branches of T, the first k, = k - n, = nR - m, branches belong to 23, and the remaining n, constitute 23,. As before, the tension across a branch b, will be denoted by q K , the flow in it by r“, and the resistance or

+ +

K

conductance of a linear resistor b,

E 23,

K

by r or g ( r * g = l), respectively. c

K

The value of a current source b, E 23, will be denoted by cK and that of the voltage source b, by eK. (We can assume, moreover, that every linear resistor has a nonnull resistance and a nonnull conductance, because a

268

Appendix 3

linear resistor of null resistance might be regarded as a voltage source of null value or be short-circuited, and a linear resistor of null conductance might be regarded as a current source of null value or be open-circuited.) D," (a = 1 , . ... m ;ti = 1 .. . , n ) and R," ( p = 1 , . . ., k ; ti = 1, ..., n) are, respectively, the primitive cutset matrix and the primitive loop matrix associated with the above-mentioned tree-cotree pair. We then have the set of equations (A3.1)-(A3.5) for P's and qK's (cf. $13 of Chapter IV):

c D&rK = O

(Kh.F.):

( a = 1 , . . ., m)

(A3.1.1)

blcE(B

or k

(Kh.F.) :

r" =p1 R/xP = 1

where x p is equal to the flow in the pth branch of Z (IC= 1, . . . , n ) ; (A3.1.2)

c R/q,

(Kh.T.):

=0

( p = 1 , . .., k)

(A3.2.1)

b,EB

or

c DKaya m

(Kh.T.):

q,

=

where ya is equal to the tension across the ath branch of Z (ti = 1, . ..,n ) ;

a= I

(A3.2.2)

gK = C"

(b,

E bc);

(A3.3)

= eK

(b,

E

%"I;

(A3.4)

YIK

K

K

gl( = g V K

(O < <

b,

q, = rt"

(0 < r < 00,

b,

(A3.5.1)

or K

E

bR).

(A3.5.2)

K

These equations are solved in the following manner. Since 8,c 2,if we denote the value of the pth current source also by 3'and that of the ath voltage source also by Go, (A3.1.2) and (A3.2.2) may be written, respectively, as

r" = c RpKxP+ kc

p= 1

q, =

IC

RkK,+pEP

for some xp's

(b,

E

p= 1

c D,"ya + c DZv+aEa

mv

nv

a= 1

a= 1

b, u bv), (A3.1.2')

for some y,'s

(b,

E

bRu Bc). (A3.2.2')

269

Theory of Linear Resistor Networks

[For b, E bc,(A3.1.2') yields nothing but (A3.3), and for b, E 23", (A3.2.2') yields nothing but (A3.4).] Substituting (A3.1.2') in (A3.5.2) and then in (A3.2.1), and taking account of (A3.4), we have

or kr

(A3.6) where

(A3.7)

(p=l,...,n,;

q=1,

..., k,).

Equation (A3.6) is called the loop equation. Since r > 0 and the rank of K

1 , . . ., k,) is k c , the matrix rqP defined in (A3.7) (which is called the loop resistance matrix) is of rank k,, and, consequently, it is nonsingular. Therefore, if the eK's and cK's (i.e., CP's) are given, we can uniquely determine the values of the xp's (p = 1 , . . . , k,) by (A3.6), and then the Y's in b, E dVu 23, by (A3.1.2'). The Y'S in b, E 23, are given by (A3.3). Thus, the flow configuration is uniquely determined. The tensions qK across b, E SjRare determined by (A3.5.2), and those across b, E 23" are given by (A3.4). The q,'s across b, E 23, are determined by means of the last equations of (A3.2. I), i.e., if b, is the pth current source, the tension across it is equal to

R,'

(b,

E b,,p

=

-

RkK,+pTK'

b,

EP) y

u

(A3.8)

P)R

Thus, we have shown a method for uniquely determining the flow configuration, as well as the tension configuration, on a linear resistor network.

270

Appendix 3

In a way dual to the foregoing, we may make use of the cutset equation: In.,

( a = 1,

..., mv),

(A3.9)

b= 1

where gab =

1

X

DXaDXbg

( a , b = 1,.

. . , mV),

bs E 5 R

(A3.10)

( a = 1, ..., m v ;

b = 1 , . .., nv),

and the matrix gab(a, b = 1, . . ., mv) is called the culsel conductance matrix. All the above equations are linear homogeneous in t", uK, cK (c'"), and eK (gb), so that it suffices to investigate the special case where only current sources or voltage sources have nonnull values. The flow and tension configurations in the general case can be obtained by the superposition of those in the special case. Let us first consider the case where there is no current source. Denoting the inverse of the loop resistance matrix rqp by rpq, we'have, from (A3.6), (A3.7), and (A3.1.2'),

(A3.11)

Now, we consider the case where there are only two voltage sources b, and b,, of value eKoand e,, , respectively, and b, is in series with a linear resistor bKl and b, is in series with a linear resistor bA1.Then, since R"' P = R:

and

RF1 = RY

for allp,

(A3.12)

we have

(A3.13)

Theory of Linear Resistor Networks

27 1

Since the matrix rp4is symmetric [see (A3.7)], it follows from (A3.13) that = atA1/aeK0. atKi/aeao

(A3.14)

This relation is called the reciprocity of linear resistor networks. In verbal terms, the reciprocity states that the flow in branch b, when there is only one voltage source of unit value placed in series with branch bAl is equal to the flow in branch bAl when there is only one voltage source of unit value placed in series with branch bK1. The relation is illustrated in Fig. A3.l(a).

(b)

Fig. A3.1. Illustration of the reciprocity.

Dually, we can prove that the tension across branch bAl when there is only one current source of unit value placed in parallel with branch bKlis equal to the tension across branch b,, when there is only one current source of unit value placed in parallel with branch b, . This relation is also called the reciprocity and is illustrated in Fig. A3.l(b). Next, we shall turn to the nonamplification property of linear resistor networks. We shall show that the tension across any branch of a linear resistor network, in which all the voltage sources as well as the current sources have zero value except for only one voltage source of value e, does not exceed e in absolute value, and, dually, that the flow in any branch of a network, in which all the sources have zero value except for only one current source of value c, does not exceed c in absolute value. Only the former case will be proved in the following since the proof of the latter can be done dually. The proof is done by explicitly writing down the formula for the q,)s in terms of r's and e. We shall denote by Rt71.::;l"c;cl I(

the subdeterminant of order kc of RPKwith rowsp = 1, . . . , kc and columns IC = K ~ .,. ., I C (as ~ ~we did in $9.6 of Chapter 111) and by R:;!;;;'$ kc,

272

Appendix 3

the subdeterminant of order k, - 1 of RpKwith rows 1, .. .,p - 1, p 1, ..., k , and columns rcZ, ..., K k C . From the definition of rqp in (A3.7), we have

+

where the summation on the right-hand side of (A3.15) is taken over all the possible combinations of k, distinct column indices of RpKcorresponding to the linear resistors (bKi E SR). [The expansion of the form (A3.15) is referred to as the Binet-Cauchy expansion in the textbooks on matrices and determinants.] The cofactor of the matrix rqpwith respect to its (4,p ) element is similarly expressed as

Dividing (A3.16) by (A3.15), we have an expression for the elements of the inverse matrix rpq. If b, is the only voltage source of nonnull value el,, then we have from (A3.5.2), (A3.1 I), (A3.15), and (A3.16),

(A3.17)

where qK is the tension across an arbitrary linear resistor. Taking into

...... 1(- 1)4-'R,KRCKz... kc

account that

q= 1

C1

Kk

]

c

(4) " ' k c 1

is the Laplace expansion of

Thus, referring to Theorem 9.16 in $9.5 and Theorem 9.17 in $9.6, we see

273

Theory of Linear Resistor Networks

that the denominator of the right-hand side of (A3.18) is equal to the sum of the products r * * r of the resistances of the branches b, , . .., bKkC

-

KkC

which form a cotree on the graph 8 , obtained from the graph 8 of the given network Jlr by open-circuiting all the current sources, whereas the numerator is equal to the sum of the products r r * . * r of the resistances K KZ

KkC

of the branches b, bKz, . .., bKkC,each term with an appropriate sign, where the sets of branches { bXz, . . ., bKk,} are chosen in such a way that b, b,, , .. ., bKkcform a cotree on 8 , and, at the same time, b, , b,, , . . . , bKkCform a cotree on 6,. Obviously, the cotrees appearing in the numerator appear also in the denominator so that the numerator cannot exceed the denominator in absolute value (note that r’s are positive). Therefore, K

we have

hKIS k a 0 l

(A3.19)

for every b, E BR.The tension across a voltage source is, by assumption, null or elo itself, so that (A3.19) holds also for b, E 23”. For the tensions across current sources, we use the following short-cut method, although it is possible to discuss them on the basis of the above formulas. Let the tension across a current source b, (of value cK = 0, by assumption) be 9, , and its increment when cKis increased from 0 to a sufficiently small value E * sgn 9, ( E > 0) be 81,. (If 4, = 0, there is no problem and we assume 9, # 0.) From the formula dual to (A3.1 l), it follows by superposition that IAq,J-+O

as E + O .

Fig. A3.2. Approximation of a current source by a linear resistor.

(A3.20)

214

Appendix 4

Let us note that the flow configuration and the tension configuration corresponding to cK= E * sgn 4, are equal to those which we shall obtain when we replace the current source bK by a linear resistor of resistance (4, + A ~ J E (cf. Theorem 16.1 in $16.2 of Chapter IV; see Fig. A3.2). Therefore, it follows from what we have just proved that

19, + AVKI 5 IeAoL

(A3.21)

from which follows, in view of (A3.20),

(A3.22) APPENDIX 4: DIGITAL AND ANALOG COMPUTATIONS

The reader who has some knowledge of computer programming will easily understand that the algorithms in this book for solving various problems concerning linear network-flow problems are presented in the form readily programmable for a digital computer. As an example, we shall exhibit a program for solving the two-terminal characteristic problems in $23, which is written in FORTRAN and is registered in the Program Library of the Computer Centre of the University of Tokyo. The names of variables used in this program do not always coincide with those we have adopted i n this book, but we believe that a program which has been debugged by many users will better serve the readers than one which, although being formally refined, might possibly contain bugs. In the following program, all the quantities, except for the variables POWER1 and POWER2 representing the values of the objective functions (D(5) and V(q), are expressed by the fixed-point (or integer) variables to get rid of the influence of rounding-off errors. Furthermore, it should be noted that the “internal branches” are to be numbered from 1 to n in the following program, whereas in Chapter V, we almost always reserved the branch b, for the “reference branch.” To use the memory space more efficiently and to remove the unessential limitation on the number of corner points of a characteristic curve, it is recommended to rewrite the program so as to replace the two-dimensional arrays JBRCHR and JBRNCH by onedimensional arrays. Furthermore, the suitable checks for input data as well as the rearrangement of the output forms (in subroutines RESULI, RESUL2, and RESUL3) might become necessary, depending on circumstances. The remarks on the usage as well as the limitations concerning the program are written on the comment cards preceding the main program, where the terms “current” and “voltage” are used for “flow” and

275

Digital and Analog Computations

“ tension ” (“ potential” being also called “ voltage ”), “ source ” and “sink” for “entrance” and “exit.” The meanings of the variables and of the subroutines used in the program are as follows (cf. the notations in $20- $23 of Chapter V) :

Simple Variables

POWER1 : POWER2 : NDNO: JBRNO : INND : JOUTND : JPRINT: INCR: I NVL :

JPRECR: JPREVL: INF:

the value of the objective function (D(5). the value of the objective function Y(q). the number of nodes in the network. the number of (internal) branches in the network. the node number, i.e., s, of the entrance node ns. the node number, i.e., t, of the exit node nt . the index for printing the intermediary results (see the comment cards preceding the main program). the entrance-to-exit flow, or the flow in the reference branch, i.e., the negative of the tension across the reference branch, or the where we assume potential at the entrance, i.e., u = [,= 0. the entrance-to-exit flow at the previous stage. the potential at the entrance at the previous stage. the large number, i.e., 999999, representing the infinity (co or 0).

r.

c,,

Subscripted Variables or Arrays

NDSTA(K) : NDEND(K) : JBRCR(K): JBRCPP(K) : JBRCPM(K) : JBRVL(K): JTHP(K) :

the node number c1 of the node nu = a+ b, which is positively incident to the Kth branch b, . the node number c1 of the node n, = a-b, which is negatively incident to the Kth branch b, . the flow 5” in the Kth branch b, . the capacity AcK in the positive direction of the Kth branch b, in regard to the incremental flow when solving the associated maximum-flow problem. the capacity -Ab“ in the negative direction of the Kth branch b, in regard to the incremental flow (see above). the tension q, across the Kth branch b, . the t, for the Kth branch b, used when solving the associated minimum-route problem (see Remark 23.2).

276

Appendix 4

the - d , for the Kth branch b, (see Remark 23.2). the number of branches ( S + n a u 6-n,) incident to the JAth node n,, i.e., the p a used in Labyrinth Algorithm 2 in Remark 20.4. the potential [, at the JAth node n,. NDVL(JA) : the ai in Labyrinth Algorithm 2, where the sign before JROUTE(1): the aiis ignored in the following program (subroutine CURINC), but the direction in which a branch lies on a conductive path is determined later when the maximum possible value of the incremental flow along the path is calculated. JBRCHR(J, K) : the numbers specifying the characteristic curve of the Kth branch, where JBCHR(1, K) is the abscissa of the leftmost semi-infinite vertical segment, JBRCHR(2, K) the ordinate of the left horizontal segment, JBRCHR(3, K) the abscissa of the middle vertical segment, JBRCHR(4, K) the ordinate of the right horizontal segment, and JBRCHR(5, K) is the abscissa of the rightmost semi-infinite vertical segment (see Fig. A4.1). JBRNCH(JW, JA): the branch number IC of the branch b, which is the JWth in order in the set 6+n, u d-n, of branches incident to the JAth node n, [if bKE6-n,, JBRNCH(JW, JA) is equal to the negative of the branch number of that branch].

JTHM(K) : JORDER(JA)

t

I

JBRCHR(4, K )

_-JBRCHR(1, K )

1

I I

= r,JBRCR(K) / JBRCHR(5, K ) JBRCHR(3, K ) JBRCH R(2, K )

Fig. A4.1. Specification of a branch characteristic.

Digital and Analog Computations

277

Subroutines RESULl, RESUL2, RESUL3 : the subroutines for printing the intermediary and final results in full or in abbreviated form. RESCAP : the subroutine for determining the J BRCPP(K)'s and JBRCPM( K)'s (i.e., the Ac"'s and -Ab"'s) corresponding to a given (compatible) pair of flow and tension configurations (see 923.2). THETA: the subroutine for determining the JTHP(K)'s and JTHM(K)'s (i.e., the 2"'s and -2,'s) corresponding to a given (compatible) pair of flow and tension configurations (see Remark 23.2). VOLINC : the subroutine for solving the associated minimum-route problem according to Eq. (23.10) in Remark 23.2, where a variant of the successive replacement (see Remark 21.8) is made use of. CURINC: the subroutine for finding a conductive path for the incremental flow according essentially to Labyrinth Algorithm 2 (see Remark 20.4) and assigning the maximum possible incremental flow along it (the associated maximum-flow problem is solved by the repeated application of this subroutine). Linear Two-Terminal Network-Flow Problem (FORTRAN Program)* Today's electronic digital computers are fast enough and powerful enough to deal with large-scale network-flow problems with sufficient accuracy in a manageable time, so that there will be no room for analog computers in practical computations. However, the discrepancy between a practical transportation or scheduling problem and its mathematical model is usually so large that only a limited accuracy is needed in solving the mathematical model. It will often be required to observe on real time how the flow or tension in/across a branch varies according to the variation of the parameters (i.e., capacities, costs, times, etc.) of some branches. On

* A reproduction of the program card deck is found on pages 280-287 at the end of this appendix.

278

Appendix 4

such occasions, an analog computer for solving network-flow problems will be of some use, or at least of “ instructive” significance. Historically, the Scott Paper Company constructed in the late 1950’s an analog computer called A L-PAC that can solve the uncapacitated Hitchcock transportation problems (cf. $23.4) with less than 25 producers and less than 108 consumers (see McCarty and Elmore [l]), and there is also a report about one for the project-scheduling problems (see Mauchly [l]). The present author once constructed a small model for instructive use (see Iri [Z]) which consisted of 14 diodes, 40 voltage sources of unit value, 10 current limiters (each with a diode connected in series) of variable value, and a patch-board with plugs and cords to interconnect these elementary branches. All the parts were those available on the market in Japan, i.e., we used ordinary gold-bond germanium diodes (SD14) for diodes, ZnNH,CI cells (UM5) for voltage sources, and the circuits of Fig. A4.4(a) containing high-frequency germanium transistors (2SA17) for current limiters. Their circuit diagrams as well as their measured characteristics

2vt 1

(a)

VL

(b)

Fig. A4.2. Circuit diagram of a diode and its measured characteristic.

(a)

(b)

Fig. A4.3. Circuit diagram of a voltage source and its measured characteristic.

279

Digital and Analog Computations

are shown in Figs. A4.2-A4.4. It will depend on the circumstances as to whether one regards these characteristic curves as good approximations of the ideal ones or not.

(4

(b)

Fig. A4.4. Circuit diagram of a current limiter and its measured characteristic.

280 C177

Appendix 4 H 1 I T C I N T W K L I N F P R TWO-TcRMIN4L

NFTWORK-FLOW

PROBLEM

C t C C

L I N F A R TWO-TFRMIN4L NFTWnRK-FLPW P R O R L F Y M4S4O 1013 KYOKO O K t I D A I R 4 12/1/66

C

C C C C C

C C

C C

C C C C

C C

C C C C

C C

C C C

C C C C C C C C

C C C

C C C

c C C C C

C C

C

c

C

C C

C C C

C C C C C C C C

C C

c C C C

C

T H I S PROGR4MME MUST 9s PROCFSSFO RY A M A C H I N F W I T H 4 MEMORY OF MORE THAN 78.70a FFFECTIVE WORDS. GFNERALITY S F C T I O N 1. T H I S PROGRAYME T R E A T S 4 G F N E R 4 L C 4 S E O F L I N E A R TWO-TERMINAL NFTWORK-FLOW PROSLEY. FOR T H F D E T A I L S O F T H E S I C N I F I C 4 N C E 6F T H E P Q O S L F H A 5 WELL P S FOR TMF R A S I C I D F A Z O F T H E S O L U T I O N A L G O R I T H M T H E USFR SHOULD R E F E R T O T H F D A D F R A NEW YETHOD O F S O L V I N G TRANSPORTATION-NFTWORK PROSLEMS T\Y H . I R I D U S L I S H E O I N THE JOURNAL O F T H E O P F R 4 T I O N S RCSFPRCH S O C I E T Y O F J ~ P A N I V O L ~ ~ r N O ~ l / 2 l O C T O P E1 9R6 0 1 . P~FS 27-n~. D F F I N I T I O N AYD S D F C I F I C b T I O N S O F T H E PROSLEM S C C T I O N 7. I N F L E C T R I C A L TFRMINOLOGY. THC P R O S L F H IS TO O R T A I N T H E VULTAGECURRENT R F L A T I O N BETWEEN A D A I R O F NODES I O Y E C A L L E D SOURCE AND THE OTHER S I N K ) O F A N O N - L I Y F A R R E S I S T O R NETWORK* E A C H O F WHOSE BRANCHES H A S A C H I R A C T E R I S T I C SUCH THAT THE R F L A T I O N BETWEEN T H E V O L T A G E ACROSS A N 0 T H F CURRFNT I N IT IS R E P R F S F N T E O B Y A MONOTONF NONO F C Q E d S I N G S T E P F U N C T I O N ON T H E PLAhlE W I T H THE CURRENT A N 0 T H E T H E PRESENT V E R S I O N O F V O L T d G F AS T H F A S S C I S S 4 4ND THF O R D I N A T F . T H F PQOGRAHHE CAN O F A L OWLY W I T H T H c S T E P F U N C T I O N S H A V I N G TWO H O Q I Z O N T A L SEGMENTS I A N 0 HENCE THREF V F R T I C A L SEGMENTSI ONE O F W H I C H GOF5 I N F I N I T E L Y UPW4RDS 4ND ANOTHER I N F I N I T E L Y OOWNWAROS). A STEP F U N C T I O N O F T H I S K I N D IS S P E C I F I E D 9 Y 4 SEOUENCE O F F I V E NUMRERSr O F W H I C H T H F F I R S T OFNOTES T H E A R S C I S S 4 OF THC L E F T M O S T V E R T I C P L SFGMFNT EXTENOINCI I N F I N I T E L Y DOWNWAR05 ( T H I S V A L U E IS SET EQUAL TO THE N F G A T I V F I N F I N I T Y I = - 9 9 9 9 9 9 9 1 I N CASE THE L E F T H O R I Z O N T A L SCGMENT F X T C N D S I N F I N I T F L Y TO T H F L F F T I . THC SFCOND DENOTES T H E O R O I N A T F O F THE L E F T 1I.F. L O W F R l H O R I Z O N T A L SEGMENT* T H E T H I R D D F Y O T F S T H F 4 R S C I S S A O F T H F M I O D L F V F R T I C A L SFGMENTI T H F F O U R T H DFNOTES T H E O R D I N 4 T F OF T H E R I G H T 1I.F. U P P E R ) H O R I Z O N T A L SEGMENT P N O T H F F I F T H DENOTFS T H F A q S C I S S A O F T H E R I G H T V E R T I C A L SEGMENT I T H I S V I L U F I S S F T FOUPL TO THF P O S I T I V F I N F I N I T Y I = 9 9 9 9 9 9 9 1 I N C 4 S F T H F R I G H T H O R I Z O N T A L SFGYFNT F X T F N D S I N F I N I T E L Y . T O T H E R I G H T I . I F T H F NUHRFR O F SFGHFIITS I5 L F S S THAN THAT M E N T I O N E D ABOVFI T H E NUM9ERS TOW4R9S T H F FN? nF T H F SFOUFNCF SHOULD R E P U T F O U A L TO T H E I N F I N I T Y I F O R FXAMPLFI 1-51017/9999999/9999999/ IS L E G A L SUT 1 - 5 1 0 1 4 IS N O T I . T H F R F S T R I C T I O N ON T H E NUMRER O F / C 1 7 / OR 1 - 5 1 - 3 / - 5 / 0 / 7 / S T C P S O F A F U N C T I O N IS NOT F S S F N T I 4 L . FOR 4 3 R 4 N C H W I T H A MORE C O M P L I C b T F O C F A R A C T E R I S T I C I 1.F. A RRANCH WHOSE C H 4 R A C T E R I S T I C IS A STEP-FtJNCTION d I T H MORF T H A N TWO H O R I Z O N T A L SEGMENTS* CAN 9 E COPPOSFO P Y C O N N C C T I N G I N S F R I F S 4ND/OR I N P A R A L L E L TWO, OR MORE. RRANCHFS W I T H S I M P L F R C H 4 R A C T E Q I S T I C S . THE MOST E S S E N T I A L C O N D I T I O N S W H l C H T H F PROCIRAMMF ASSUMFS A Q F AS FOLLOWS. F I R S T . THE C H d R d C T F R I S T l C S T E P F U N C T I O N O F 4NY SRANCH SHOULD H A V E A V F R T I C A L SFGMENT W I T H N U L L A B S C I S S A OR P H O R I Z O N T A L SEGMENT P A S S I N G T H E ORIGIN. T F C O N D I T H F R F SHOULD F X I S T P V O L T 4 G E C O N F I G U R A T I O N ( I . € . A S F T O F V A L U F S OF RR4NCH V O L T A G F S I S P T I S F Y I N G 4 L L T H E RRANCH C H I R A C T F R I S T I C S I N P A I R W I T H THE N U L L CURRENT C O N F I G U R A T I O N 1I.E. HOWEVERI T H F SET OF SRANCH CURRFNTS F A C H R E I Y G FQUAL TO ZERO). T H F S E C O N D I T I O N S PRE S A T I S F I F O SY ALMOST A L L T H E P R O B L E M WE 1 Y P A R T I C U L A R . MAXIYUM-FLUW PROBLEMS. D Q O I N A R I L Y FNCOUNTEQ. T Q 4 N S P O R T A T I O N P R O S L F M S t SHOQTCST-ROUTE PROSLFMSe C R I T I C I L - P A T H S C H E D U L I N G PRORLEMSI ETC.. I N O P E R A T I O N S RESEARCH S 4 T I S F Y T H E S E C O N D I T I O N S ( S E E S F C T I O N S 7. R. 9 4 N 0 101. S F C T I O N 1. R E L A T I O N TO Y I N I M l Z 4 T I O Y P R O B L E Y S T H E CURRFNT C O N F I G U R d T I O N 1I.F. T H F S F T OF V A L U E S O F BRANCH CURRENTS1 CORRESPONDING T O P C F Q T 4 I N V d L U E O F T H E I N P U T CURRENT 11.E. T H F O V F R 4 L L SOURCF-TO-SINK CURRENT1 COMPUTED B Y T H I S PROGRAMME IS T H E ONF TH4T. S A T I S F Y I N G THF C O N T I N U I T Y C O N D I T I O N O F CURRENTS 1I.E. T H E CURRENT LAW O F K I R C H H O F F I W I T H T H A T I N P U T CURRENT, M I N I M I Z F S T H E P R I M 4 L POWCRv WHFRE THC P R I M A L POWER A S S O C I A T E D W I T H 4 CURRFNT C O N F I G U R A T I O N IS n E F I N E D AS T H F SUM OF T H E RRANCH P R I M A L POWFRS ANT) T H F RRANCH P R I M A L POWER OF 4 BRANCH IS A F U N C T I O N I O F THE CURRENT I N T H 4 T PRANCH) FROH W H I C H T H F F U N C T I O N O F THE B R 4 N C H C H A R A C T F R I S T I C I W I T H THE CUQRENT PEGAROEO A S T H E I N D E P E N D E N T V A R I A B L E PND T H E V O L T d G E A S T H F D E P F Y D E N T l IS O B T A I N E D BY OIFFERFNTIbTION.

NTWKO000 NTWKOOOI NTWKOOO~ N T W K 0 005 NTWKOLO6 NTWKOb07 NTWKOOOB NTWK0009 NTWKOOIO NTWK0013 NTWK0016 NTWK0019 NTWKOOZ2 NTWKOO25 NTWKOO28 NTWK0031 NTWKOO34 NTWKOd37 NTWK0040 NTWKO043 NTWK0046 NTWK004C NTWKOO52 NTWKOG55 NTWK0058 VTWK0061 VTWK0064 NTWK0067 NTWK0070 NTWK0073 NTWKO076 NTWK0079 NTWKOO82 NTWKOO85 NTWKOO88 NTWK0091 NTWK0094 NTWK0097 NTWKO100 NTh’KOlOJ NTWKOlO6 NTWKOlO9 NTWKOll2 NTWKO115 NTWKO118 NTWKOl2l NTWKO 1 2 4 NTWKOlZ7 NTWK0130 NTWK0133 NTWK0136 NTWK0139 NTWKO142 NTWKO145 NTWK0148 NTWKOl51 NTWK0154 NTWK0157 NTWKO 160 NTWKOl63 NTWKO166 NTWK0169 NTWK0172 NTWK0175 NTWKO178 NTWKOl8l NTWK0184 NTWKO 1 8 7 NTWKO190 NTWK0193 NTWK0196 NTWK0199 NTWK0202 NTWK0205 NTWKO208

Digital and Analog Computations C C C C C C C C C C C

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T H E V O L T A G F C O N F I G U R A T I O N 1I.F. T H E SET O F V A L U E S O F BRANCH VOLTAGCS) CORRFSPONOINC TO A C F R T A I N V A L U E O F T H E I N P U T V O L T A G E 1I.E. T H E P O T E N T I A L D I F F E R E N C E BETWFEN T H E SOURCE NODE AND THE S I N K NODF) COMPUTE0 B Y THIS PROGQAMYF IS THE ONE THAT, S T A T I S F Y I N G THE T H E V O L T A G E LAW O F K I R C H H O F F I C O N T I N U I T Y C O N D I T I O N O F V O L T A G E S 1I.E. W I T H THAT I N P U T VOLTAGE. L I N I Y I Z F S T H E D U A L POWER. WHERE T H E D U A L DOWFR A S S O C I A T E D W I T H A V O L T A G C C O N F I G U R A T I O N IS D E F I N E D A S T H E SUM OF T H E SRANCH D U A L POWFRS AND THE RQbNCH O V A L POWER O F A RRANCH IS A F U N C T I O N ( O F T H E VOLTAGE O F THAT R R A N C H I FROM W H I C H T H E F U N C T I O N OF THC BRANCH C H b R A C T E R I S T I C [ W I T H T H F V O L T A G E AS THE I N D E P E N D E N T V A R I A R L F AYD T H E CUQRENT A S THE D F P E N D E N T I 1 5 O B T A I N E D 3 Y OrFFCRFNTlbTION.

NTWKO211 NTWKO2 14 NTWKO217 NTWK0220 NTWKO223 NTWK0226 NTWKO229 NTWKO232 NTWK0235 NTWK0238 NTWK0241 NTWKO244 NTWK0247 S F C T I O N L. NTWK0250 O U T L I N F O F THE C O M P U T A T I O N A L MFTHOD NTWK0253 T H I S DROGRAMME S T A R T S W I T H THF N U L L CURRFNT C O N F I G U R A T I O N AND COMPUTES T H E VOLTAGE C O N F I G U R A T I O N AND T H E CURRENT C O N F I G U R A T I O N NTWK0256 A L T E R N A T E L Y I N SUCH A WAY T H A T THE I N P U T V O L T A G E OR T H E I N P U T NTWK0259 NTWKO262 CURRENT I N C R E A S E S I N EVCQY STAG=. R C S D F C T I V E L Y I U N T I L EITHER THE NTWK0265 I N P U T VOLTAGE OR THE I N P U T CURRFNT IS NOT I N C R E A S E D ANY YORE* I.€. U N T I L T H E F N O O F T H E F F A S I B L E R F G I O N IS REACHED. THE I C U R R E N T AND/ NTWKO268 OD V O L T A G E ) C O N F I G U R A T I O N CV?RFSPONOING TO AN I N T E R M E D I A T E VALUE O F N T W K O 2 7 1 T H E 1NClJT CURRFNT OR V O L T A G E I I N T E R V F O I A T F SETWEEN TWO COMPUTED NTWKO274 NTWK0277 R F S U L T S I MAY BE O R T A I N F D B Y T H F L I N E A R I N T F R P O L A T I O N . F X C F P T FOR T H E P R I M A L POWER A N 0 T H E O U A L DOWERI A L L T H E V A R I A B L E S NTWKO28O NTWKO283 I N T H I S PROGRAMME ARE O F I N T E G E R TYPF. SO T H A T I T IS NECESSARY TO I N T R O D U C F A P P R O P R I A T F S C A L E F A C T O P S =OR 9 E S P E C T I V E Q U A N T I T I E S . NTWKO286 NTWK0289 SFCTION 5. D A T A CARDS AND OUTDUT NTWKO292 NTWK0295 T H E DATA CARDS FOR T H I S PROGRAMME SHOULD B E ARRANGE0 A S FOLLOWS. NTWKO298 ON THF F I R S T CARD, THE NUVRER O F YODFSI THE NULBER O F PRANCHESI T H F NODF NUMqER O F T H E SOURCE NODFI T H F NODE NUMRER O F T H E S I N K NOOF R T W K O 3 0 1 NTWKO304 AND T H F I N O F X FOR P R I N T I N G AQE PUNCHED I N T H I S ORDER R Y F O R M A T 1 4 1 8 r ? o X ~ I l l . I F O N L Y T H F C 0 4 F I G U Q A T I O N AT T H E F I N A L STAGE O F T H E NTWK0307 NTWK0310 C O ~ D U T A T 1 r ) N IS N F E D F D THE I N O E X FOQ D R I N T I N G IS S E T EQUAL T O 0 ( O R MAY B E L F F T B L P N K I . I F . I N A O O I T I O N I THE P R I M A L AND T H E D U A L POWER NTWK3313 O F I N T F Q P E D I A R Y S T A G E S O F T H E COMPUTATION ARE NEEDED I T IS SET EQUAL N T W K 0 3 1 6 TO 1 . AND I F A L L T H F CURQENT A W VOLTAGE C 9 Y F I G U R A T I O N S AT THE NTWK0319 THE NTWKO322 I N T ~ R M C D I A R YS T A G F S ARE NEEDED I T IS SET E Q U A L TO 2. NTWK0325 COMPUTATION IS C A R R I E D OUT WORF E F F I C I E N T L Y I F THE NODES ARE NUMRERFO I N T H F ASCFNDINCI OQDER FROY T H E S I N K T O T H E SOURCF I W H I C H I NTWKO328 HOWEVFR. IS NOT O R L I G A T O Q Y I . NTWK0331 ON T H E S U C C E E D I N G CAROSr THE BRANCH NUMBER. THE NODE NUMBER O F THE N T W K 0 3 3 4 NOOF FROM W H I C H T H F P R I N C H STARTS, T H F NODF NUMSER OF THE NODE AT NTWK0337 W H I C H T H F RRANCH F N n S AND T H F SFQUCNCE O F F I V E I N T E G E R S W H I C H NTWK0340 NTWKO343 S D C t I F Y THC C H A R A C T F P I S T I C OF T H F SRANCH A Q F PUNCHED I N THIS ORDER RY F O R M A T I R I R I . F A C H CARD C O N T A I N S T H F I N F O R M A T I O N S FOR ONE BQANCH. N T W K 0 3 4 6 THF OROCR O F C a R D S AND T H b T O F RRANCH NIJMPFRS MAY NOT N F C E S S A R I L Y P F N T W K 0 3 4 9 C O I N C I D E N T SO LONG AS THE BRANCHFS ARF NUM'IERED WITHOUT O U P L I C A T I O N N T W K O 3 5 2 OR VACANCY. NTWK0355 T H E 4 9 0 V c SLOCK O F CAROS I I T S NUMPFR IS F Q U A L TO THE NUMPER O F NTWK0358 'IRANCHFS P L U S I1 C O N S T I T U T E S THE NECFSSARY AND S U F F I C I E N T D A T A FOR A N T W K 0 3 6 1 PRO'ILFM. I F ONE WbNT5 T O S P L V F MORE THAN ONE PRORLEM I N A RUN, ONE NTWK0364 MAY MERFLY S T A C K T H E B L O C K S O F DATA CARDS FOR THOSE PROBLEMS. NTWK0367 NTWK0368 I T IS RECOMMENDFD T O AOD A RLANK CART) AT THE E N 0 OF THF D A T A CAQD DECK. NTWK0369 T H E F O L L O W I N G IS AN F X A M P L E O F A RLOCK O F D A T A CAROS FOR A PROBLEV N T W K 0 3 7 0 W I T H 6 NOOFS AND 7 PRANCHFS. NTWK0373 0376 7 5 1 2 0379 5 1 5 4 -1 -1 0 1 1 0382 2 5 ? -1 -7 0 ? 2 0385 3 4 2 -1 -1 0 1 1 0388 4 4 % -1 -7 0 1 1 0391 7 a -1 -1 0 2 1 0394 5 6 3 1 -1 -2 0 4 1 0397 7 2 1 -1 -4 3 2 2 0400

0403 0404

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28 1

S F C T I O N 6. R F S T Q I C T I O N S O F T H E DRFSCMT V E R S I O N OF T H F PROGRAMMF THE PQOGRbMPF M b K F S NO CHFOUF FOR T H F Y O N O T O N I C I T Y OF T H F RRANCH C H A R A C T C D I S T I C S NOR FOR THE D U D L I C A T I W OR THE VACANCY O F NODE AND QDANCH N U M R ~ Q 5 , YOR FOQ qTHER K I N D S O F I L L C G I T I M A C Y O F PRO'ILFM SPFCIFICATIOYS. T H r R F S I I L T Z FOR THF I L L E G A L D A T A ARE UNOEFINED. R F S I D F S THE R E S T R I C T I O Y S W F Y T I O N F O I N S F C T I O N 2 9 THE PRESENT V F R S I O N O F T H F P R O t R A V M F DOES NOT 4 D V I T T H F PROSLEM W l T H YORE THAN

NTWKO406 NTWK0409 NTWK0412 NTWK0415 NTWK0418 NTWK0421 NTWKO424

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Appendix 4

1000 RRANCHES OR W I T H WORE THAN 501) NODES. ( B U T . T H I S MAY B E F R E E D FOOM @Y S I M P L Y R E W R I T I N G T H C D l M F N S l O N STATEMENTS A T THE B E G I N N I N G O F T H E M 4 1 N PROGR4MME I F T H F MFMORY C A P A C I T Y O F T H E P V A I L A R L E MORFOVFR. THE 4AXIMUM NUMRER O F RRANCHES CnMPUTcR 1 5 LARGF.1 (HOWFVFR. WF CAN F R E E I M C I O F N T TO A N O n F IS L I M I T C O W I T H I N 20. O U P 5 E L V C 5 FROM T H I S L I M l T 4 T I O N SY I N T R O D U C I N G F I C T I T I O U S NODES AND "FtANCHFS I N MOST CAqES.1

NTWK0427 NTWK0430 NTWK0433 NTWK0436 NTWK0439 NTWKO442 NTWK0445 NTWK0448 q F C T l O N 7. A P P L I C A T I O N TO A MAXIMUM-FLOW PRO3LEM NTWK0451 THE SFOIIFNCE OF F I V E NUMBFPS R E P R C S F N T I N G THE C H A R A C T F R I S T I C OF A NTWK0454 m9ANCH I S T O R E PUT EOUAL TO T H F N F G A T I V E OF THE C A P A C I T Y O F THE NTWK0457 ROUTE I N THE N F G A T I V E O l O E C T l O N . 0 9 THC C A P A C I T Y I N THE P O S I T I V E NTWK0460 DIRECTIONI I N F I N I T Y l = 9 9 0 9 9 0 9 1 AN0 I N F I N I T Y ( = 9 9 9 9 9 9 9 1 . THE I N D E X NTWK0463 THE I N P U T CURRENT I N THE F I N A L C O N F I G U R A T I O N N T W K 0 4 6 6 FOR P R I N T I N G MAY B E 0. 1 5 EOUAL TO T H E MAXIMUM F C A S I R L E FLOW FROY THE SOURCE TO THE S I N K , NTWKO469 AYO T H F BRANCH CURRENTS T O T H E F L W S I N THE R E S P E C T I V E ROUTES. NTWK0472 THOSE FIRbNCHcS WHOSF V O L T 4 G c S ARF FOUPL TO 9 9 9 9 9 9 9 (OR - 9 9 9 9 9 9 9 1 NTWK0475 r n N S T l T U T F T H F CUT-SFT '0 T H F Y I N I M U M CUT V 4 L U E 4NO A L L O F THEM ARE NTWK0478 SATURATCD. NTWK0481 NTWKO484 4 P P L I C A T l O N TO b YINIMUM-COST T R A N S P O R T A T I O N PRORLEM SFCTION 0 . NTWKO487 THE CURRFNT I N A RRANCH I5 COVPARcT) TO T H F FLOW OF COMMODITY I N NTWK0490 CORRESPONOIYG ROUTC. THC P R I M A L POWFR I S K W A L TO T H F T O T A L COST. NTWKO493 T H F SEQ'IFNCF O F F I V E NUMSERS R F P R E S N T I N G T H E C H A R A C T E R I S T I C O F A NTWK0496 RPANCH 1 5 TO E F P U T EOUAL TO T H E N E G A T I V E OF THE C A P A C I T Y O F THE NTWK0499 ROUTE I N THE N F G A T I V E D I P F C T I O N . THE N F G A T I V E O F T H E COST PER U N I T NTWK0502 NTWKO505 FLOW I N THE N E G A T I V F D I R C C T T O N v Z F R O t THE COST PER U N I T FLOW I N THE P O S I T I V F O I R E C T I O N ANO THE C 4 P A C I T Y I N THE P O S I T I V E D I R E C T I O N . THE NTWK0503 I N P I I T CURRCNT CORRcSPONOC TO THE T O T A L FLOW THROUGH T H F TRPNSPORTANTYK0511 I F T H F M 0 5 T FCONOMICAL NTWK0514 T l n N NFT#OOC FQOM THE SOIIRCF TO T H F q I N K . W4YS OF T R 4 N S P O R T I N G COYMOOITY 4RE NFFOFD FOR V A R I O U S V A L U E S O F NTWK0517 I F THE NTWKO520 TOTAL FLOW, T H F I N D E X FOO P R I N T I N G IS TO R F S F T F Q U 4 L TO 2. VALUE O F T H F T O T A L FLOW IS F l X E n TO P G I V E N VALUE, I T IS RFCOMYENOEO NTWKO523 TO AUGMFNT T H F G l V F N NFTWORK W I T H A F I C T I T I O U S SOURCE AND A NTWK0526 F I C T 1 T l r ) U S ROUTE. W H I C H L 4 T T C R . S T A R T I N G F 9 0 M T H F F I C T I T I O U S SOURCF N T W K 0 5 2 9 A N 9 E N O I N G AT T H F R F A L S O U R C F I HAS T H F N U L L C I P A C I T Y I N T H F N E G A T I V F N T W K 0 5 3 2 D I P F C T I O N . THE C A P A C I T Y I N THE P O S I T I V E D I R E C T I O N EQUAL TO THE G I V E N N T W K 0 5 3 5 NTWK053R TOTAL FLOW V A L U E AN0 THE N U L L COST, ANO THFN TO SOLVE THE PRORLEM FOR THF AUGMFNTEO NETWOQK W I T H THE I N D F X FOR P R I N T I N G SET FQUAL TO O.NTWK0541 NTWK0544 S F C T l O N 0. 4 P P L I C A T I O N TO A SHORTF
-). P I 9 J o 9 C P P ( ] " " I ) JERCPY . UFTlr 1 O l ' / c k ' C I P ' v ~ * i , \ T e l I r " O ) .hlO.)CVOI I T "JnFtCR(1" 1 0 0 ) J O O V L ( I 'I^ I .JTUP( 1 n-l .JTul'( 1'D)lr J3(iFQ(S n - 1 .EIOVL( C"c) 9 NFT'.' 2 ~ J O r ) U T ~ ~ ~ ~ O ~ ~ k ~ n c L ~ * ~ ~ l r J ~ ~ ~ ~ A ~ ~ ~ ~ l ~ NET*# J 0 9 C2 ~ ~ ~ ~ ~ l " ~ ~ ~ ~ J n 1I

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Digital and Analog Computations

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Appendix 4

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Digital and Analog Computations

28 5

C b L L P F S U L ? l I N V L i P O ~ ' ~ D 2 ~ ~ O S T D ~ ~ ~ c N ~ ~ J P R V L . N D V L ~ ~ R 9 N O RES3 ~ N D N11 O~2l RES3 1 2 CND R E 5 3 13

on ~ E T U D N

SUQROUTINF R F S C 4 P l J ~ R N O ~ J Q Q C Q ~ J n ~ V L ~ J P R C H R ~ J R R C P P ~ J S R C P V l . nlUFhlq?nh( J U R C P I I O L O l ~ J R D V L l 1 3 ~ ~ l ~ J ~ 9 C P ~ l l ~ O ~ l ~ J ~ R C lJ9RCHRlc,lD90l 00 1 9 r r l i J Q R N 0 IFlJSPV~1KI-JQoCHDI7tu)) 7 * 1 ~ 7 1 JSRCPPl~)=J~RCH1l1~Y)-JSOCRIKl J 9 ~ C P ~ l ~ l ~ J P R C R l ~ I - J a o C H R o GO TO 1 1 7 I F l J 9 D V L l K I - J Q R C H 9 l 4 r Y ) l 4.9.4 .1 J S Q C P P l Y l = J ~ R C H R l 5 ~ K ) - J R 9 C R O JBRCPI.IlKl=JRRCRlKl-J~QCHl?l1~K~ GO TO 11 4 JRRCDPIYI-0 JSSCPvlYI=O 13 CONTINUC RClUDN

CND

RESC 1 PRESC V l l ~ 02O l ~ RESC 2 1 RESC 3 RESC 4 RESC 5 RESC b RFSC 7 REX 8 RESC 9 R E X 10 RESC 11 RESC 12 RESC 13 RESC 14 RESC 1 5 RESC 16

SUPROUTINF T H F T A l J P R N O ~ J 9 R C S . J 9 ~ V L ~ J R 9 C W Q ~ J T ~ P ~ J T H ~ ~ I ~ F Tl H E 1 O l V C N ~ l O YJ ~ R C Q l l O C O l r J R Q V L ~ I " r ~ l ~ J T H P l l ~ ~ ~ l ~ J T H ~ l l ~ O W n El T ~ 1JRRCHRl6r1000l THE1 THE1 DO 14 K = l r J Q R N O IFIJRRCRIKI-JRRCHRlIlr)) 7r1.7 THE1 THE1 1 JTHPlKl=JRRCH912~Kl JTHYCYI=INF THE1 GO 10 14 THE1 THFT 7 IFIJR9CRlKI-JRRCHRI1rYI) 4.114 1 JTHPIYl=JRQCHQl4.Kl THE1 JTWMIK)=-JPRCHRI?.KI THE1 GO i n 14 THE1 4 I ~ l J S R C R l K ) - J 9 R C H R l + t Y I l 6.596 THE1 THE1 C JTHDIKIrINF JTH~lKl=-JRRCHR14~Kl THE1 GO i n 14 THE1 6 JTHDIYI=JQRVLIKI THE1 THET JTHMlKlr-JRRVLIKI 14 C O N T I N U F DFTOPN THE1 THET END

1 2 21

3 4

5 6

7 n 9 10 11 12 13 14 15 16 17

THET is

19 20

S U 9 R O I l T I N F V O L I N C ~ N ~ N ~ ~ I N F I I N F ~ ~ J Q R ' I O ~ J ~ R V L ~ N ~ V L . J O R D ICt R ~ JV S OR L IN C 1 H 1YDSTA~NDFND~JTHP~JTHV~l~VL~I~YOeJ~UT~~~LARFLl VOLI 2 O l V F N S l O N J P R V L l 1 0 0 D l ~ Y O V L l ~ ~ ? l ~ J O R ~ F R ~ 5 ~ ~ l ~ ~ ~V ~O LTI A l3 l ~ ~ ~ l ~ 4 I N ~ F N ~ l 1 0 0 0 ) r J T H P l 1 0 O O ~ ~ J T H M l l C O ~ l ~ L 8 S F L l 5 O G l ~ J Q R N C H l 7 O ~ 5 O OV lO L I VOLI 5 90 1 5 J A = I * N D N O VOLI 6 NDVLlJAIrINF V O L I 7 L A S F L IJ A l = O VOLI 8 15 CONTINUC VOLI 9 N D V L I JOUTNDI=O V O L I 10 LPSFLIJOUTNI)l=l V O L I 11 1 JC=O V O L I 12 DO 16 JA=l.NDNO VOLI 13 N*JOQDFQI J A I V O L I 14 00 I7 11l.N VOLI 1 5 K=JPPNCHIIrJAI V O L l 16 I F I K I lr9.2 V O L I 17 2 JB=YDcNDlKI V O L I 18 J=NOVLIJAI+JTHMIYI V O L I 19 no i n 4 V O L I 20 3 K=-K V O L I 21 JS-NOSTPIKI V O L I 22 J=NI)VLIJAI+JTHPIKI V O L I 23 4 IFIJ-ItJ'?I 6.695 V O L I 24 J=IYF voii 6 I F I N D V L I J R I - J I 17~1717 V O L I 26 'I N D V L I J P I = J V O L I 27 JC= 1 V O L I 28 L A R F L I J9I-1 V O L I 29 17 CONT?NUF V O L l 30 16 CONTINUE VOLI 3 1 IFIJCI 9rEil

is

286 Q

00 1 4 K = I . J P R N O JWl-Yn5T41 K l JWZ=NDFND(Kl

JRRVLlK)=NnVLIJWII-NnVL(JW?I 10

r n w i ~ c ~ ~ INVL=NnVLIINND) RFTUPN FY0

Appendix 4 VOLI VOLI VOLl VOLI VOLl VOLI VOLI VOLI

32 33 34 35

3b 37 38 39

CH U ~R l 1 SUFIROUTINE C U R I N C l N O N O ~ J ~ R C R ~ J R O U T F ~ J R R S T A I L A R F L . J O R O F R ~ J B R N C CURl 2 ~JRRCPP~J~RCPM~NOSTA~N~END.JCIP1INCR~INF~INNO~JOUTNO~ CUR1 3 01YCNSION J R 0 U T F ( 5 0 0 ~ ~ J R R 5 T A 1 5 O O ) . L P R E L ( S O C ) . J O R O E R ~ 5 O O ~ ~ CO UO R lO ~ ~4 IJ9RCRl1000)~J~RCPPl10~0~rJRRCPMl1300~rN~5TAllOOO~~NDENDll CURI 41 7JBRNCHl70~500) CURI 5 DO 1 0 J 4 = 1 . N O N O CURI 6 JROOTFl.J4l=O CURI 7 JRRSTA( 1 4 ) s O CURI 8 L4UFLlJA)=O CURl 9 10 C O N T I N U P C U R l 10 J R W T F I 1 I = INNO C U R I 11 LA9~LlIYYQ)=l CURI 1 2 J=l CURI 13 60 T0 2 C U R I 14 1 J9QST4lJ)=I J CURI 1 5 . = J +.l JROUTFlJ)=LRJ CURS 16 LA9FLILRJ)sl C U R l 17 CURl 1 8 IF(LQJ-JOUTN0) 7.797 7 JRJ=J'?flUTF(JI C U R l 19 N=JQRST41J) CURI 2 0 = JOQOFR I J R J I CURI 2 1 lill=hl+l CURI 2 2 CURI 2 2 1 I F I N I - Y ) 15.15.16 CURl 222 1C. 0 0 70 I = N I v M YrJRONCHI I .JRJ) CURl 2 3 CURI 2 4 I F ( < ) 4~4.3 1 LCAP=JP'?CPPIK) CURI 25

Digital and Analog Computations

287

____ LSJ=YVFYDlKl

4

* 6 20 16

7

9

10 11 71

19 17

77

14

CURI CURI CURl CURI CUR1 CURI CURI CURI CURI CURI CURI CURI CURI CURI. CURI CURI CURI CURI CURI CURI CURI CURI CURI CURI CURI CURI CURI CURl CURI CURI CUR1 CURI CURI CURI CURI CURI CURI CURI

26 27 28 29 30

31 32

33 34

35 36 37 38 39

40 41

42 43 44

45 46 47

48 49

50 51 52 53

54 55 56 57 58

59 60 61 62 63

288

Appendix 5

APPENDIX 5: REDUCIBILITY OF A LINEAR PROGRAMMING PROBLEM TO A LINEAR NETWORK-FLOW PROBLEM

Linear network-flow problems (hereafter abbreviated as LNF) constitute a special subclass of the class of linear programming problems (hereafter abbreviated as LP). As we have seen, more details are theoretically studied about LNF than about LP, and much more efficient practical procedures are available for LNF than for LP. Therefore, it is useful to investigate under what kinds of conditions an LP is reduced to an LNF. The set of branches 23 of the network JV of an LNF may be assumed to consist of voltage sources, current sources, and diodes (see $13.3 of Chapter IV). We shall denote by Bv,23, ,and 8 , the set of voltage sources, current sources, and diodes, respectively, where B = dvu Bcu 8 , . Without loss in generality, we may assume that ?BVis loop-free and 23, is cutset-free, and choose a tree-cotree pair on the graph 8 of JV such that 2 2 BV and z 2 23, (cf. Appendix 3). DKa(a = 1 , ..., m ; IC = 1, . . . , iz) and RPK( p = 1 , . . . , k ; K = 1, . . , n) denote the primitive cutset matrix and the primitive loop matrix associated with (2,T). The flow in branch b, is denoted by 5" and the tension across b, by q , , and, especially if b, is a current source or a voltage source, its value is denoted by cK or e , , respectively. Then, the minimization problem in terms of flows, i.e., Problem I, as defined in $15 of Chapter IV, is written as

(2,z) .

(A5. 1)

whereas the minimization problem in terms of tensions, i.e., Problem 11, is written as I

I

(A5.2)

Reducibility of a Linear Programming Problem

289

where we artificially add the minus sign before each q K , as well as eKand c", in order to make the second inequality in (A5.2) similar in form to that in (A5.1). On the other hand, a general LP may be expressed in the standard form as follows (see, e.g., Dantzig [l]): I

I

C aKixK= gi

(i = I,

.. ., r),

":I

x"L0

(ICEKp),

(A5.3)

where the X"'S (IC= 1, . ..,n) are variables, the a,"s (i = 1, . . .,r ; K = 1, ...,n) form a constant matrix of rank r (hence, r 5 n), the g"s form a constant vector, the pK's form another constant vector, and K p is a subset of (1, ..., n}. Moreover, we may assume that p K= 0

for every

I C EK p ,

(A5.4)

for, if p K # 0 for a certain IC E K p , then we may modify the expressions of the problem by introducing a new variable X" (whose sign is not restricted), as well as a new equality constraint

and replacing the term pnX" in the objective function by p KX" to expel the sign-restricted variable X" from the objective function. [If we admit as a unit branch of an LNF the series connection of a voltage source and a diode (in the case of Problem I) or the parallel connection of a current source and a diode (in the case of Problem 11), then the same 5" or q K will appear in both the second and the third relations of (A5.1) or (A5.2). Thus, this kind of modification of the problem corresponds to regarding a series connection or a parallel connection as one branch in an LNF.] Now let us define the "equivalence" between two LP's. An LP with aKi,gi, p K and another LP with CKi, FK are defined to be equivalent to each other if and only if there exists a nonsingular matrix Tii (iJ = 1, ..., r ) and a matrix SKA( K , I = 1,. . . n), which is the product of a

a',

290

Appendix 5

permutation matrix and a diagonal matrix with positive diagonal elements, such that otii=

TJiStiaii,j

( i = 1, . . ., r ;

K = 1,.

.. , n),

(A5.6)

j = l A=I

( i = 1, . . . , r ) ,

(K=

1, . . . , n ) .

(A5.7) (A5.8)

The role of the matrix T,' is to replace the system of equations in (A5.3) by an equivalent system, whereas that of the SKais to renumber and to rescale the X~ variables. This definition of equivalence has a practical significance in that as soon as a solution (i.e., a set of values P)of one of the equivalent problems is found, the solution of any other problem can be obtained simply by permutation and rescaling of the RK's. Furthermore, we shall say that an LP is reducible to a linear network-flow problem with flow variables [hereafter abbreviated as LNF(F)] when there is a network for which Problem I is equivalent to the given LP, and that an LP is reducible to a linear network-flow problem with tension variables [to be abbreviated as LNF(T)] when the former is equivalent to the Problem I1 for some network. Since the coefficient matrix of the system of equations in (A5.1) is obtained by deleting the columns corresponding to the branches of d, from the primitive cutset matrix of the original graph 8,it is equal to a primitive cutset matrix of the graph 8, obtained by open-circuiting the branches of Bc of the original graph 6 (Theorem 9.16, 99.5, Chapter 111). Similarly, the coefficient matrix of the system of equations in (A5.2) is a primitive loop matrix of the graph (5, obtained by short-circuiting the branches of B, of the original graph Q (Theorem 9.13). Hence each of them consists of O's, l's, and - 1's only and contains as a submatrix the unit matrix of order, respectively, m or k. Furthermore, if we choose an arbitrary sct of n z or k independent columns and perform the row operation which transforms the submatrix constituted from those columns into the unit matrix of order m or k, then we have the primitive cutset matrix of (Gc or the primitive loop matrix of (5, associated with another tree-cotree pair. It is easy to transform the matrix aKiin (A5.3), by means of an appropriate T,' and an appropriate permutation matrix &', into the basic form

Reducibility of a Linear Programming Problem

29 1

such as

so that we shall assume that the coefficient matrix of the equality constraints of the given LP is already in the form (A5.9). Then, for the given LP to be reducible to an LNF(F) or LNF(T), it is necessary and sufficient that the matrix (A5.9) should become a primitive cutset or loop matrix of a graph by multiplying each row as well as each column by a suitable positive number. It is at least necessary that it is possible to choose a set of positive numbers I , , . . . , I,, p,+,, ...,p, such that

(A5.10)

where f: consists of 0, I, and - 1 only. If such a set cannot be found, the given LP is not reducible to an LNF. If such a set is found, then, writing the matrix and vectors obtained after the transformations (A5.6)-(A5.8) as a,’ [equal to the right-hand side matrix of (A5.10)], gi, and p K , we should search for a graph having the aKias a primitive cutset matrix. If such a graph cannot be found, the given LP is not reducible to an LNF(F). If such a graph is found, we construct a network based on that graph by connecting in parallel with each of the branches (which form a tree) corresponding to the columns of a unit submatrix of aKithe current source of value g’(where i is the row number in which the corresponding column has 1) and making the branches corresponding to the IC’S in K p the diodes and the other branches the voltage sources of values p , . The LNF(F) on

292

Appendix 5

the network thus obtained is equivalent to the given LP. Then we search for a graph having the uKi as a primitive loop matrix. If such a graph cannot be found, the given LP is not reducible to an LNF(T). If such a graph is found, we may construct a network based on that graph by inserting in series with each of the branches (which form a cotree) corresponding to the columns of a unit submatrix of uKithe voltage source of value -gi and making the branches corresponding to the IC’S in K p the diodes and the other branches the current sources of values p K. The LNF(T) on the network thus obtained is equivalent to the given LP. This process affords a means to determine whether or not a given LP is reducible to an LNF, if we have a practical algorithm for solving the problem of determining the A’s and the p’s as well as an algorithm for the problem of constructing a graph from a given primitive cutset or loop matrix. The latter problem is one of the most fundamental problems in the field of topological synthesis of networks (mentioned also in Remark 12.3 in 912 of Chapter IlI), and various methods for solving it were recently proposed. However, among them, those which seem to be both theoretically complete and practically efficient are too complicated to expound here. The interested readers may see Iri [3] and Tutte [l]. The former problem, i.e., the problem of determining the Ai’s and p i s (i = 1, . . ., r ; j = r 1, . . ., n) so as to satisfy (A5.10), is easy to attack. The condition (A5.10) is written as

+

2.1.. = I.-p. I IJ 1J J

where 1,

= 0,

(i = 1, ..., r ; j = r

+ 1, . ..,n),

1, or - 1, or as Izi = lZijlpj

for all i, j

such that I , # 0.

(A5 11)

Since obviously the l i ’ s and p i s are determined only up to a common factor, we may set 1, = 1 or p l = 1. (A5.12) Starting from (A5.12), we repeat procedures (i) and (ii) alternately: (i) Let j , be a column for which the pjo is not determined yet. If there are rows i such that lijo# 0 and that the 1;s are already determined, then determine the p j o by pjo

= 1i/llijol.

(A5.13)

If different i’s give different values to pjo, i.e., if (A5.13) is inconsistent, then conclude that the solution is nonexistent. (ii) Let i, be a row for which Aio is not determined yet. If there are rows j such that lioj # 0 and that the p j 7 s are already determined, then

293

Reducibility of a Linear Programming Problem

determine the lioby li,

= P-j

Iliojl

(A5.14)

.

If (A5.14) is inconsistent, then conclude that the solution is nonexistent. Thus, we shall have all the Ai’s as well as p i s determined, or conclude that the solution does not exist, or encounter the case (see Fig. A5.1) where,

L

Fig. A5.1. Separable matrix.

if we denote by I (or J ) the set of i’s (or,j’s) for which the 2;’s (or p i s ) are determined and by I (or J ) the set of the remaining i’s (or,j’s), then we have

I 11. . = O

if i E I

and

orif

j E J

i E i

and

jEJ.

(A5.15)

In this case, the matrix L is separated into two submatrices, one with rows I and columns J and the other with rows I and columns J , for the former of which all the Ai’s and p j ’ s are determined. Hence, we may continue the procedure similar t o the foregoing for the latter submatrix, starting by putting for an arbitrary i in I li= I (A5.16) or for an arbitraryj in J. pj = 1

EXAMPLE.Let us consider the following LP:

+ 3x6 + 6x6

X1

2x2 x3

- x6 - x7

=

+ 3x6 + 3x7 - x8

= 15,

x6-4x7+

= 1;

+ x4

X1

x3

+ 2x9 = 10, + 2x8 + 4x9 = 14,

-x5-

x8

xl, x2, x3, x4, x6, x8 2 0; xi

+ 6x3 + x 5 + 15x6 + 6x7 + 6 x 9 4 min.

I, (A5.17)

294

Appendix 5

The equality constraints in (A5.17) are rearranged into a basic form as follows :

; 0 0

0 0 0

Hi 0 0

:I:

8

i-1

o i o

1 0

1

j

O

jl E.

8

-1 0 3 - 1 - 2 3 - 1 0

(A5.18)

Then the 2:s (i = 1, . . ., 5 ) and the p;s ( j = 6, ..., 9 ) are determined consistently as shown in (A5.19). Hence, with xl, x2, 3x3,x4, x5, 3x6, 3x7,

?; 1 1

5 1 1

-

3

5

1

t

3 3 -1 0 0

0 0

0 1 0 -1 -1

2 2 0 -2 0

-1 3 3

(A5. 19)

xB,2 x 9 as the new variables, the given LP is rewritten in the following form :

0

0

0

0

0

1 1 -1 0 0

1;

0 0 1 0 1 1 0 0 -1 1 -1 -1 1 - 1 0

(A5.20)

3x3, x4, 3x6, x8 2 0;

xl,

x2,

x1

+ 2 - 3x3 + x 5 + 5 - 3x6 + 2

3x7

+3

*

2x9-tmin.

There is no graph which contains the matrix shown in (A5.20) as a primitive loop matrix, so that the given LP is not reducible to an LNF(T), whereas

Reducibility of a Linear Programming Problem

295

Fig. A5.2. Graph for the LP of (A5.17).

we have the graph of Fig. A5.2 which contains that matrix as the primitive cutset matrix associated with the tree indicated by bold lines in the figure. Assigning voltage sources, current sources, and diodes to the branches of this graph according to the afore-said rule, we have the network of Fig. A5.3, the LNF(F) of which is equivalent to the given LP. Hereit should be noted that the variables xl, 3x3, and 3x6 appear both in the inequality constraints and in the objective function, so that the corresponding branches are the series connections of a voltage source and a diode.

Fig. A5.3. LNF for (A5.17).

296

Appendix 6

APPENDIX 6: RELATED PROBLEMS AND EXTENSIONS OF NETWORK-FLOW PROBLEMS

The following is a list, with brief comments, of problems which are of interest in connection with network-flow problems, but which are not developed further in this book. (1) Maximum dynamicJlow through a network: In $20 of Chapter V, we dealt with the problem of maximizing the entrance-to-exit flow through a two-terminal network in which the capacities in the positive and the negative directions are defined for each branch. Let us suppose, furthermore, that for each branch, the time required to pass it is defined, and consider the problem of sending as much commodity as possible from the entrance ns to the exit nt within a given time T, where commodity is allowed to be held over some time at a node and each branch is assumed to allow the flow only in the positive direction. This kind of transportation problem was called the problem of maximal dynamic3ow and was given an effective solution by Ford and Fulkerson. From the given network Jlr with of the capacity and time characteristics, we construct the new network 1 kind treated in $20 as follows, where % and b denote the set of nodes and branches of Jlr, whereas % and 3 denote those of J.We quantize the time to have the significant instants z = 0, 1, 2, .. . , T. Corresponding to each node n, of %, % has the set of nodes n,,o, n,,,, .. ., nu,=. If 23 has a branch b, such that d’b, = n, and a-b, = n8, that the capacity in the positive direction is cK (20) (that in the negative direction is null), and that the time required to pass it in the positive direction is u, (ZO), then 8 has the set of branches which connect na,* to ns,T+,,, (z = 0, 1, ..., T and z + u, 5 T ) with the capacity in the positive direction cK and the null capacity in the negative direction. In addition, 3 has the branches connecting to for all c1 and for z = 0, 1, ..., T - 1, each with the infinite capacity in the positive direction and the null capacity in the negative direction. The flows in the latter kind of branches correspond to the holding-over of commodity at the nodes. Our problem is then converted into the single maximum-flow problem on the network with ns,o as the entrance and nt, as the exit. Ford and Fulkerson established a method to construct a solution of the maximum-flow problem on J by solving the minimum-cost transportation problem on Jlr where the u,’s are interpreted as the unit costs for branches in N.For the details of their method, see Ford and Fulkerson [l].

Related Problems

297

(2) Quadratic network-pow problems: A linear network-flow problem is solved far easier than a general linear programming problem of the same size because of the simplicity of the constraint relations among variables i.e., because we can make full use of its topological structure. What about a quadratic network-flow problem, where the network contains also linear resistors, or, in general, a branch characteristic curve may consist not only of vertical and horizontal segments but also of straight line segments of arbitrary positive gradient? As is obvious, an arbitrary nonlinear characteristic curve can be better approximated by a piecewise linear function than by a step function. A quadratic network-flow problem is a special case of a quadratic programming problem, whereas a linear network-flow problem is a special case of a linear programming problem. For solving a quadratic programming problem, Wolfe’s method and Beale’s method as well as their modifications are well known (see Kunzi and Krelle [l]). Here arises a question as to whether and to what extent the topological structure of a network can be utilized to devise a simpler algorithm for quadratic network-flow problems than those known for the general quadratic programming problems. The solution of a linear network-flow problem is reduced to the solutions of maximum-flow problems and those of minimum-route problems as was shown in Chapter V, whereas the solution of a quadratic network-flow problem requires the solutions of the problems of linear resistor networks as well. However, these solutions correspond to the solutions of simultaneous linear equations with the loop resistance matrix or the cuset conductance matrix as the coefficient matrix (see Dennis [l] and Katzenelson [l]; cf. also Appendix 3). Thus, unlike the case of linear network-flow problems, additions and subtractions alone are not sufficient, but a large number of multiplications and divisions are also needed in solving quadratic network-flow problems. In summary, it seems that a quadratic network-flow problem is not so remarkably easier to treat than a general quadratic programming problem as is a linear network-flow problem than a general linear programming problem. (3) Analysis and synthesis of terminal-pair capacities of a symmetric network: The problem mentioned in $20.4 of Chapter V, i.e., the problem of clarifying the properties of terminal-pair capacities of a network, was given an elegant answer by R. E. Gomory and T. C. Hu [l] in the case of symmetric networks. By a symmetric network, we mean a network .N each of whose branches is provided with the capacities equal in value in the positive direction and in the negative direction. If, as we did in $20.4,

298

Appendix 6

we denote by capthe maximum possible value of the total entrance-to-exit flow through .Af with its node na as the entrance and np as the exit, and if we set can= co for all a, (A6.1) then we have cap= cpa for all 01 and p (A6.2) from the symmetry of the network and ~"'"3

- min(c""2 2

,caZa3) for all al,a2 ,and a3,

(A6.3)

By analysis of the terminal-pair capacities of a symmetric network, we mean the problem of determining cap's of a given network, and by the synthesis the problem of constructing a network (i.e., determining the capacities of its branches) when all of its terminal-pair capacities cap's are given. The results obtained by Gomory and Hu are as follows (see for the details, Ford and Fulkerson [I13 or Berge and Ghouila-Houri [l]): (i) Among m(m + 1)/2 terminal-pair capacities cap's of a connected network with m + 1 nodes, at most m can have mutually distinct values. (ii) In order to determine all the cap's, we need not solve m(m 1)/2 maximum-flow problems for all the possible pairs of nodes, but we have only to solve m maximum-flow problems. (Gomory and Hu gave the practical algorithm for determining cap's in this way.) (iii) The inequalities (A6.3) are not only necessary for a set of values cap's to be the terminal-pair capacities of a symmetric network, but they are also sufficient.

+

Moreover, they elaborated a simple algorithm for constructing a network whose terminal-pair capacities are not less than (arbitrarily) prescribed values [not necessarily satisfying (A6.3)] and the sum of whose branch capacities is as small as possible. Unfortunately, however, Gomory and Hu's method seems to depend strongly upon the symmetry of the network, so that it is difficult to generalize many results obtained for the symmetric networks to the case of asymmetric ones. The generalizations to the case of asymmetric networks proposed are so far only partial. (In this regard, see, e.g., Mayeda [l].)

(4) Networks with branch gains and losses: In the foregoing, we regarded a branch as a route through which something flows without increasing or decreasing its magnitude. However, in some practical cases, e.g., in electric power transmission lines, in water pipes, etc., a branch or a route is leaky, i.e., some part of flow entering it leaks out, and only the

299

Related Problems

remaining part is conveyed through it. Moreover, there are many problems, as we shall show, which are formulated in terms of networks containing the leaky branches or those of a similar kind. In general, we define a network with branch gains and losses as a network N ( 0 ; C,, . .., Cn). Here, 0 is a graph, C, is the branch characteristic of branch b, (as defined in $13.3 of Chapter IV) with which two convex functions (p,(?), t,P(q,) are defined (where 5" and q, are the flow in and the tension across b,) and is a matrix with rows corresponding to nodes na and columns corresponding to branches b, such that

z;

a

-

-

9; # 0

only if na = a'b,

or na = 8-b,.

(A6.4)

9 may thus be regarded as an extended incidence matrix of the graph 0 in the sense that it is obtained from the incidence matrix 5 of 0 by replacing the 1's and - 1's of the latter matrix by arbitrary numbers. Furthermore, we assume that the flows in a network with branch gains and losses satisfy the continuity condition

and the tensions satisfy (A6.6) where M is the number of nodes and are appropriately chosen numbers (which are the generalization of node potentials) (cf. $13.1 and $13.2). Intuitively, we may regard a branch b, of a network of this kind as a branch, of the same kind as we dealt with in the foregoing, carrying flow 5" and tension q, , which, however, is not connected directly to its endnodes but has the transformers of turn ratios and -ZKs (where ab, = na - nI in 6)inserted between its ends and the nodes (nu and ns) of the graph 0 (see Fig. A6.1).

z/

Fig. A6.1. Typical branch in a netwmk with branch gains and losses.

300

Appendix 6

The minimization problem in terms of flows, which is the extension of Problem I in $15.1 of Chapter IV, is to obtain tK’s, which, satisfying (A6.5), minimize (A6.7) and the minimization problem in terms of tensions, which is the extension of Problem 11 in $15.2, is to obtain qK’s(as well as (:s), which, satisfying (A6.6), minimize

The equilibrium problem in terms of flows and tensions, which is the extension of Problem I11 in $15.3 is to obtain F’s and q,’s (as well as re’s) such that the 5“’s satisfy (A6.5) and the qK’s(A6.6) and that

(t‘, q,)

E C,

for all

K.

(A6.9)

The relations among these three problems can be established as the natural extensions of those expounded in $15.4. However, the flow or tension configuration [satisfying (A6.5) or (A6.6)] on a network of this extended kind no longer forms a loop or a cutset on the graph 6. One of the most remarkable peculiarities that is not seen in an ordinary network is the existence of the so-called “source loops” and “ sink loops,” such as shown in Fig. A6.2, around which flows are generated and absorbed.

Fig. A6.2. Examples of (a) sink and (b) source loops.

It has been pointed out that the basic theorems on the maximum-flow problem in $20 of Chapter V can partly be extended to this case (see, e.g., Mayeda and Van Valkenburg [l]) and that the algorithms developed in $23 can be extended mutatis mutandis to the networks with branch gains and losses whose branch characteristics are step functions (see

Related Problems

301

Jewel1 [l] and Iri et al. [l]). However, like the case of quadratic networkflow problems mentioned in (2), the special character of the network, i.e., its dependence on a graph, is not so much useful for simplifying the solution algorithms in this extended case as it is in the case of ordinary networks. As a final comment on the network problems with branch gains and losses, the so-called machine-loading problem will be formulated. We have to produce m kinds of items on n machines, where it takes h,"+' of time and it costs as much as ea,m + to produce one unit of item a on machine j (a= 1, .. .,m ;j = 1, . ., n). If it is required to produce ca units of item a (a= 1, . ..,m) and machine j is available in the duration not longer than cm + j( j = 1, . .,n), then what is the best plan of allocating the productions of items to machines, i.e., what (?*'"+j's minimize

.

.

(A6.10)

under the conditions 5"p

2 0,

m+n

1 t", = c"

(A6.11)

(a= 1, ..., m),

(A6.12)

( P = m + l , ..., r n + n ) ,

(A6.13)

,=mi 1

m

ChtrflscS a= 1

where 50L,m+j is the number of units of item a to be produced on machine j ? Thus, the problem is mathematically the same as the Hitchcock transportation problem (investigated in $23.4.3) except for the existence of nonunity coefficients h t ' s in (A6.13). The machine-loading problem is formulated in the network problem shown in Fig. A6.3. If the solution exists, it is determined by calculating the flow configuration which, satisfying all the branch characteristics with a certain tension configuration, maximizes the flow entering the entrance (not necessarily equal to the flow going out of the exit). The algorithm for calculating such a configuration is established as a natural extension of that for the Hitchcock problem. For the details, see Iri et al. [l]. ( 5 ) Networks with bundle constraints: Another extension of the network-flow problem is the problem of a network with bundle constraints, by which we mean the problem that takes account of the capacity (or span) or, more generally, cost associated with the sum of the flows in (or the sum of the tensions across) the branches belonging to a subset called a

302

Appendix 6

303

Related Problems

bundle in addition to the ordinary capacities, spans, and costs associated with single branches. A bundle is called aflow bundle if the sum of the flows in the branches belonging to it is in question, and it is called a tension bundle if the sum of tensions is in question. In the following, we shall consider two kinds of networks with bundle constraints, i.e., networks with flow bundles and those with tension bundles. Let us begin with a network N = N ( 0 ;C1,. . ., C,; Bl, B,, . . .; f

f

C1, C, f

f

, ...) with flow bundles where 0 and C, are, as before, the graph

f

and the characteristics of the branches, whereas the B;s are the flow J

bundles and the CI)s are the characteristic curves (of the same kind as f

those defined in $13.3) concerning the bundles. We denote by r" and q, the flow and the tension in/across the branch b, . Furthermore, we introduce the notations 5' and q i to denote the concepts to be called the flow and the f

f

tension of the flow bundle ?Bi. Specifically, we set f

ti= f

1 5". b,E%

f

If we denote by I' = [b', c'] the projection of C ito the f

f

f

t

axis, then the

f

capacity restrictions for the bundle flows are written as bi 5 ti4 ci I

f

.

(i = 1, 2, . .).

f

The problem for Jlr which corresponds to Problem I in $15.1 is then to f

obtain 5"s and Y's that minimize f

(A6.14)

under the continuity conditions of the branch flows,

C D,"5"

=0

(U

= 1,

...,m),

(A6.15)

K

and those of the branch flows and the bundle flows, (A6.16) f

as well as under the capacity restrictions

b" 5 5" 5 c",

b' f

5 ti5 ci, f

f

(A6.17)

304

Appendix 6

where D," is a cutset matrix of the graph 6, qKor qi one of the convex f

functions associated with C, or C i , and I" = [b", cK] the projection of C, f

to the 5 axis. The problem corresponding to Problem I1 in $15.2 and that corresponding to Problem 111 in $15.3 can be derived from the foregoing problem, in the same way as we handled the proof to Theorem 15.3 by means of the Farkas theorem, as follows; the former problem is to obtain q,'s and q { s f

that minimize ~(II, q) = C +,(v") f

+

,

c P(qi) i

f

(A6.18)

f

under the continuity conditions for the tensions now taking the form

C Rpl(ij, = 0

( p = 1, .. ., k),

(A6.19)

X

(A6.20) f

as well as the span restrictions

(A6.21)

di 5 qi 5 ei

d, 2 q K 2 eK

f

f f where RPKis a loop matrix of 6, and J, = [d,, e,] or

Ji= [di,ei] the f

I f

projection of C, or C i to the q axis; and the latter problem is to obtain

lK's, ti's, f

f

qi)s, and qK's such that they satisfy the continuity conditions f

(A6.15), (A6.16), (A6.19), (A6.20), and at the same time the characteristics of the branches and the bundles: (A6.22) ~1,) E C, (ti,qi) E Ci * (5.9

7

f f

f

The problems concerning a network with tension bundles are formulated in the way dual to the foregoing, where the continuity conditions take the following form: 1D,"p = 0, (A6.23) K

r" = 5" +

c

ti,

(A6.24)

Bisb, t

(A6.25) (A6.26)

305

Related Problems

The concept of a flow bundle in its simplest form was introduced under the name “ faisceau” by French researchers (see Berge and Ghouila-Houri [l]), whereas the concept of a tension bundle was utilized by W. S. Jewell [2] to deal with a job which may be performed at more than one place in the arrow diagram. In this connection, it seems necessary to point out that the concept of homolog branches which was also introduced by French researchers (see Berge and Ghoulia-Houri [l]) can be regarded as a special case of a tension bundle. A set of homolog branches is defined as such a set that all the branches in it are required to carry one and the same amount of flow with one another. However, this condition of homologness is equivalent to assuming in mutually disjoint tension bundles bithat (“ = 0 t

for the branches b, in a bi, regarding the F’s as the flows in the branches. t

In fact, then, (A6.23) becomes the ordinary continuity condition for flows and (A6.24) states that the flows in the branches of a certain tension bundle bi are equal to the bundle flow t i ,thus securing the homologness. t

t

The network problems with bundle constraints cannot be solved in a purely graphical procedure, unlike the ordinary network-flow problems, but they require in general something like a simplex method for the general linear-programming problems. This will not be surprising if we observe that the class of network problems that are provided at the same time with branch gains/losses and with bundle constraints is equivalent to the class of all the linear-programming problems (as will be seen without much difficulty). The extension of problem is done, in general, at the cost of simplicity and practicality of solution procedure! However, Jewell [2] showed that the problem for a network that has a tension bundle alone and in which only the span restrictions are in question (ie., a feasible tension configuration is to be required) can be reduced to a linear network-flow problem, and T. Kobayashi [l] pointed out that, by the dual technique, the problem for a network that has a flow bundle alone and in which only the capacity restrictions are in question can be reduced also to a linear network-flow problem. (6) Multicomrnodityflow: Thus far, we have been concerned with the flows of a single kind. A natural extension is suggested to increase the number of kinds of flows, i.e., to consider many kinds of commodities to be transported through a network in which the capacity restriction is imposed on the sum of the amounts of these kinds of flows in each branch. If necessary, the costs will also be considered. The network problem of this extended kind is called the multicommodity network-$ow problem, in

306

Appendix 6

contrast with the ordinary single-commodity network-flow problem. Actually, many practical transportation problems are to be formulated as multicommodity network-flow problems. However, to our regret, we have no practical algorithms for multicommodity problems so simple and effective as those for single-commodity problems, nor does it seem that much has been investigated theoretically about the particular features other than those shared by the general linear-programming problems. The only thing that we might remember when we are forced to deal with multicommodity problems would be the decomposition technique, i.e., the formulation in generalized linear programming (see Dantzig [l]), by which we can make use of the single-commodity network-flow techniques to solve the subproblems under a master linear-programming problem (see Ford and Fulkerson [2]). We have also an excellent summary by W. S . Jewel1 [3] of the history of the researches concerning network-flow problems and their extensions.

The following list is not intended to be a comprehensive or exhaustive one by any means, for it contains only those papers and monographs which are explicitly referred to in the text. Rather comprehensive lists of literature in the related fields will be found in Berge and Ghouila-Houri [l], Ford and Fulkerson 111, and Jewel1 131. BELLMAN, R. [I] “Dynamic Programming.” Princeton Univ. Press, Princeton, New Jersey, 1957. [2] “Applied Dynamic Programming.” Princeton Univ. Press, Princeton, New Jersey, 1962. BELLMAN, R., and KARUSH, W. [l] Mathematical programming and the maximum transform. J. Soc. Znd. Appl. Math. 10, No. 3, 550-567 (1962). BERGE,C. [l] “Thtorie des Graphes et Ses Applications.” Dunod, Paris, 1958. English transl.: “The Theory of Graphs and its Applications” (transl. by A. Doig). Methuen, London, 1962. BERGE,C., and GHOUILA-HOURI, A. [l] “Programmes, Jeux et R h a u x de Transport.” Dunod, Paris, 1962. In English: “Programming, Games and Transportation Networks ” (transl. by M. Merrington and C. Ramanujacharyulu). Methuen, London, 1965. DANTZIG, G. B. [l] “Linear Programming and Extensions.” Princeton Univ. Press, Princeton, New Jersey, 1963. DENNIS,J. B. [l] “ Mathematical Programming and Electrical Networks.” MIT Technol. Press, Cambridge, Massachusetts and Wiley, New York, 1959.

307

308

References

FARKAS, J. [l] Theorie der einfachen Ungleichungen. J. rein. angew. Math. 124, 1-27 (1902). FORD, L. R., Jr., and FULKERSON, D. R. [l] “Flows in Networks.” Princeton Univ. Press, Princeton, New Jersey, 1962. [2] A suggested computation for maximal multi-commodity network flows. Management Sci. 5 , No. 1,97-101 (1958). GOMORY, R. E., and Hu, T. C. [l] Synthesis of a communication network. -J. SOC.Ind. Appl. Marh. 12, No. 2, 348-369 (1964). IRI, M. [l] Supplements to the theory of trees, cotrees, multi-trees and multi-cotrees. In “RAAG Memoirs of the Unifying Study of Basic Problems in Engineering and Physical Sciences by Means of Geometry” (K. Kondo, ed.), Vol. 3, A-X, pp. 4-22 (1962). Gakujutsu Bunken Fukyukai, Tokyo. [2] An analogue computer for network-flow problems. Paper presented at 10th Ann. Intern. Meeting, August, 1963, Inst. of Management Sci., Tokyo (1963). J1. [3] On the synthesis of loop and cutset matrices and the related problems. In “RAAG Memoirs of the Unifying Study of Basic Problems in Engineering and Physical Sciences by Means of Geometry” (K. Kondo, ed.), Vol. 4, A-XIII, pp. 4-38 (1968). Gakujutsu Bunken Fukyukai, Tokyo. [4] A critical review of the matroid-theoretical and the algebraic-topological theory of networks. In “RAAG Memoirs of the Unifying Study of Basic Problems in Engineering and Physical Sciences by Means of Geometry” (K. Kondo, ed.), Vol. 4, A-XIV, pp. 39-46 (1968). Gakujutsu Bunken Fukyukai, Tokyo. M. IRI, M., AMARI,S., and TAKATA, [l] Algebraical and topological theory and methods of linear programming with weak graphical representation. In “RAAG Memoirs of the Unifying Study of Basic Problems in Engineering and Physical Sciences by Means of Geometry” (K. Kondo, ed.), Vol. 4, G-IX, pp. 421-464 (1968). Gakujutsu Bunken Fukyukai, Tokyo. W. S. JEWELL, [l] Optimal flow with gains. Operations Res. 10, No. 4, 476-422 (1962). [2] Divisible activities in critical path analysis. Operations Res. 13, No. 5, 747-760 (1965). [3] A primal-dual multi-commodity flow algorithm. Rept. ORC 66-24, Operations Res. Center, Univ. of California, Berkeley, California, 1966. KATZENELSON, J. [l] An algorithm for solving nonlinear resistor networks. Be// System Tech. J . 44, NO. 8, 1605-1620 (1965). T. KOBAYASHI, [l] On maximal flow problem in a transportation network with a bundle. J. Operations Res. SOC.Japan 10, No. 314, 69-75 (1968). KONIG,D. [I ] “Theorie der endlichen und unendlichen Graphen.” Akademische Verlagsgesellschaft, Leipzig, 1936; Chelsea, New York, 1950.

References

309

KONDO,K., and IRI,M. [l] On the theory of trees, cotrees, multi-trees and multi-cotrees. In “RAAG Memoirs of the Unifying Study of Basic Problems in Engineering and Physical Sciences by Means of Geometry” (K. Kondo, ed.), Vol. 2, A-VII, pp. 42-83 (1958). Gakujutsu Bunken Fukyukai, Tokyo. W. KUNZI,H. P., and KRELLE, [l] “Nichtlineare Programmierung.” Springer, Berlin, 1962. In English: “Nonlinear Programming” (transl. by F. Levin). Ginn (Blaisdell), Boston, 1966. R. KURATA, [l] Primal dual method of parametric programming and Iri’s theory on network flow problems. J. Operations Res. SOC.Japan 7,No. 3, 104-144 (1965). KURATOWSKI, C. [l] Sur le problkme des courbes gauches en Topologie. FundamenfaMath. 15,271283 (1930). MCCARTY, C. E., and ELMORE, W. C. [l] The application of a unique analog computer to product distribution. TAPPZ (Tech. Assoc. Pulp and Paper Znd.) 43, No. 4, 192A-195A (1960). MAUCHLY, J. W. [l] Critical-path scheduling. Chem. Eng. 69, No. 8, 139-154 (1962). MAYEDA, W. [l] On oriented communication nets. IRE Trans. Circuit Theory CT-9, 261-267 (1962). MAYEDA, W., and VANVALKENBURG, M. E. [l] Properties of lossy communication nets. ZEEE Trans. Circuit Theory CT-12, 334-338 (1965). MINTY,G. J. [l] Monotone networks. Proc. Roy. SOC.A257, 194-212 (1960). [2] On the axiomatic foundations of the theories of directed linear graphs, electrical networks and network-programming. J. Math. Mech. 15, 485-520 (1966). ORE,0. [l] “ Theory of Graphs.” Am. Math. SOC.,Colloquium Publ., Vol. 38, 1962. ROCKAFELLER, R. T. [l] Convex programming and systems of elementary monotonic relations. J. Math. Anal. Appl. 19, 543-564 (1967). TUTTE,W. T. [l] From matrices to graphs. Can. J. Math. 16, 108-127 (1964). WHITNEY, H. [l] Non-separable and planar graphs. Trans. Am. Math. SOC.34, 339-362 (1932). [2] Planar graphs. Fundamenta Math. 21,73-84 (1933). [3] 2-Isomorphic graphs. Am. J. Math. 55,245-254 (1933). [4] On the abstract properties of linear dependence. Am. J. Math. 57, 509-533 (1935). ZOUTENDUK, G. [l] “Methods of Feasible Directions.” American Elsevier, New York, 1960.

Index

A Additional branch, 154, 159 Additional node, 154 Admissible direction, 15 Admissible pair, of flow and tension, 78 Admissible set, see Matching (set) Amari, S., 301,308 Analog computer, 277-279 Antizigzag rule, 143,181 Approximation problem, 1 16-120 Arrow diagram, 3, 191,238 construction from partial order, 240-243 Articulation point, 68 cutting at, 68 Assignment problem, 214-216,252-255 solution algorithm for, 235-238 Associated maximum-flow problem, see Maximum-flow problem Associated minimum-route problem, see Minimum-route problem

B Bases for cutsets, 54-58 for loops, 54-58 transformation of, 57,58 Bellman, R., 24, 188, 307 Berge, C., vi, 28, 151, 298, 305, 307 Birkhoff, G., 158 Boys-and-girls problem, 157 Branch, 29, 75 Branch characteristic, 75, 78-87 of parallel connection, 80-82 of series connection, 83, 84

Branch gains, 298-302, 305 Branch losses, 298-302,305 Bundle, 301,303-305, see also Flow bundle, Tension bundle C

Capacity, 13 1 residual, 135 terminal-pair, 150, 297, 298 Carrier, 36 negative, 36 positive, 36 Characteristic curve, see Convex function, Branch characteristic Closed path, 42 algorithm for examining existence of, 42 elementary, 43 Compatible pair, of ffow and tension configurations, 78 Complement subgraph, 32 Complete pentagon, 74 Concave function, 12 strictly, 12 Conductivity of branch, 135, 179, 195 of path, 135 Conjugate convex functions, 21 Connected components, 38 algorithm for decomposing graph into, 41 Connectedness of graph, 38 of nodes, 37 Continuity condition, see Kirchhoffs laws

311

312

Index

Convex function, 11-26 characteristic curve of, 19 continuity of, 13 definition of, 11, 12 derivative of, 15 differentiability of, 15 with one variable, 18-26 strictly, 12 Convex hull, 10 Convex set, 8-11 Convexity, of solutions, 101 Cotree, 48-54 complementarity with trees, 5 1 definition of, 48 Covering set, 160 Crash duration, 239 Critical path, 192, 239 Cut, 133 Cutset, 36 elementary, 37, 45 integral, 36 minimal, 37,44,45 orthogonality with loops, 47 simple, 36 Cutset conductance matrix, 270 Cutset equation, 270 Cutset matrix, 54, see also Primitive cutset matrix Cutset-free subset, 36, 49 Current limiter, 84, 279 Current source, 78

D Dantzig, G. B., 289, 306, 307 Dennis, J. B., vii, 297, 307 Digital computer, 274-277, 280-287 Dilworth, R. P., 168 Dilworth decomposition, see Partially ordered set Diode, 79,80, 278 Distance, 177 terminal-pair, 189 Doig, A., 307 Doubly stochastic matrix, 158 decomposition into permutation matrices, 158

Dual graphs, 73, 192 Dual sentences, 66 Duality, 66, 67, 87, 192 Dummy job, 240

E Egervhy, J., 159 Electric network, see Resistor network Elementary closed path, see Closed path Elementary cutset, see Cutset Elementary loop, see Loop Elementary path, see Path Elementary segregation, see Segregation Elmore, W. C., 278,309 Entrance, 110, 131 Equilibrium problem, 89, 300 Equisignum decomposition, 263-267 theorem of, 264 Equisignum inclusion, 264 Equivalence transformations, of networks, 120-1 28 Event, 191, 238 Event time, 238 Existence, of solutions, 94, 108-110, 146, 147, 176, 186 Exit, 110, 131

F Factor, 173 Faisceau, 305 Farkas, J., 257,264, 308 Farkas’ theorem, 256-263 Feasible configuration, 88, 89 Flow, 75, 76 Flow configuration, 76 Flow bundle, 303, 305 Ford, L. R., Jr., vi, 133, 141, 151, 200, 249, 296, 298, 306, 307, 308 Fulkerson, D. R., vi, 133, 141, 151, 200, 249, 296, 298, 306, 307, 308 Fundamental set, see Primitive set

G Ghouila-Houri, A., vi, 151,298,305,307 Gomory, R. E., 151, 297,298, 308

313

Index Graph, 28 definition of, 28-30 transformations of, 58-61

H Hall, P., 159 Hitchcock (transportation) problem, 212-214, 255, 301, see also Transportation problem capacitated, 212 equivalence with general transportation problems, 216, 217 solution algorithm for, 221-226 Homolog branches, 305 Horizontal segment, 24 Hu, T. C., 151, 297,298, 308

I Incidence matrix, 33,63 rank of, 40 reduced, 41, 63 Incidence relation, 29 Inner product, 32 Integrity, of solutions, 147, 181,202,203 Internal branch, 111 Interval, 11 closed, 11 open, 11 Intrinsic branch, 154, 159 Intrinsic node, 154 Iri, M., 28, 74, 278, 301, 308, 309 Isomorphic graphs, 3 1 Isolated node, 31

Kirchhoffs laws, 6,76,77 Kobayashi, T., 305, 308 Konig, D., 28, 159,308 Kondo, K., vii, 28,308, 309 Krelle, W.,297, 309 Kiinzi, H. P., 297, 309 Kuhn-Tucker theorem, 89 Kurata, R., 255, 309 Kuratowski, C., 74, 309 Kuratowski graphs, 74

L Labyrinth algorithm 1, 136, 137, 146 Labyrinth algorithm 2, 145, 146 Levin, F., 309 Linear graph, see Graph Linear programming problem, 129,305 reducibility to linear network-flow problem, 288-295 in standard form, 289 Linear resistor, 80 Loop, 36 elementary, 37, 43 integral, 36 minimal, 37, 44,45 orthogonality with cutsets, 47 simple, 36 Loop equation, 269 Loop-free subset, 36,49 Loop matrix, 54, see also Primitive loop matrix Loop resistance matrix, 269

M J Jewell, W.S., 301,305,306,307,308 Job, 191,238 Job time, 191,238

K b N S h , w., 24,307 Katzenelson, J., 297, 308

McCarty, C. E., 278,309 Machine-loadingproblem, 301,302 Matching (set), 159 Matching problem, 159-171 algorithm for, 162 Matching theorem, 161, 165 Mauchly, J. W.,278,309 Maximum-flow minimum-cut theorem, 133

3 14

Index

Maximum-flow problem, 130-150,193 associated, 196, 200 solution algorithms for, 134-150 Maximum dynamic flow, 296 Maximum-separation minimum-route theorem, 177 Maximum transform, 23, 24, 82, see also Convex function Mayeda, W., 298,300,309 Menger, K., 159 Merrington, M., 307 Milestone, 240 Minimal cutset, see Cutset Minimal loop, see Loop Minimal vector, see Vector Minimization problem, 87-101 in terms of flows, 88,288,300 in terms of tensions, 88,288, 300 Minimum (point), 27 global, 27 local, 26 Minimum convolute, 82 Minimum-route problem, 174-188, 194 associated, 196, 200 solution algorithms for, 178-188 Minty, G. J., vii, 67, 309 Mizoo, Y., vii Module, see Vector space Multicommodity problem, 305, 306

N Network, 75 Network-flow problem, 87 general solution algorithms for, 243-249 linear, 129 nonlinear, 249-252 quadratic, 297 reducibility of linear-programming problem to, 288-295 Neumann, I. von, 158 Node, 29 Node capacity, 131 Nonamplification, 103-106, 271-274

Nonnegative loop-span condition, 176, 186, see also Existence of solutions Nonseparable components, 70 Nonseparable graph, 70 Normal duration, 239 Nullity, of graph, 4 5 4 8 , 50 0

Objective function, 88 Open-circuiting, 3 1 Optimal configuration, 88, 89 Ore, O., 28, 309 Out-of-kilter method, 249

P Parallel, 30, 63-66 Partial order, among jobs, 191 Partially ordered set, 168 Dilworth decomposition of, 168-171 Path, 37, see also Closed path elementary, 37 Planar graph, 73 Potential, 76 Primitive cutset matrix, 50,61, 288 Primitive loop matrix, 50,61,288 Primitive set of cutsets, 50 of loops, 50 Principle of optimality, 188 Problem I, see Minimization problem Problem 11, see Minimization problem Problem 111, see Equilibrium problem

Q Quadratic programming, 267,297

R Ramanujacharyulu, C., 307 Rank, of graph, 45-48, 50 Reciprocity, 106, 107, 271 Reduced incidence matrix, see Incidence matrix

315

Index Reference branch, 111 Reorientation, 3 1 Residual capacity, see Capacity Residual span, see Span Resistor network linear, 267-274 nonlinear, 5-7, 89 Rockafeller, R. T., vii, 309 Route, 177 S

Saturation, of branch, 134 Saturation pattern, 143 Scheduling problem, 3-5,88, 191 minimal-cost, 238-243 Segregation, 43, 44 elementary, 44,45 Self-loop, 3 1 Sensitivity, 252 Separable graph, 69 Separable matrix, 69 Separation, 175 Series, 63-66 Short-circuiting, 31 Shortest-path problem, 190 Simple cutset, see Cutset Simple loop, see Loop Simple vector, see Vector Simultaneous replacement, 188 Sink, see Exit Sink loop, 300 Source, see Entrance Source loop, 300 Span, 175 residual, 179 Star-to-mesh transformation, 127, 128 Subgraph, 32 Subgraph problem, 171-174 Successive replacement, 188 Sunaga, T., vii Supply-demand problem, 153-158 Symmetric network, 297, 298 System of distinct representatives, 167, 168

T Takata, M., 301, 308 Tenseness, of branch, 178, 179 Tension, 75,76 Tension bundle, 303, 305 Tension configuration, 77 Terminal-pair capacity, see Capacity Terminal-pair distance, see Distance Thomsen graph, 74 Transportation problem, 1-3, 88, 209217, see also Hitchcock (transportation) problem of empty ships, 21 1 with given total cost, 210 with given total flow, 210 multientrance-multiexit, 21 1, 255 Tree, 48-54 complementarity with cotrees, 5 1 definition of, 48 Triangular inequalities, 150-153, 188-1 90 2-isomorphic graphs, 70, 71 2-isomorphism, 70, 71, 121 invariant properties in regard to, 70 Two-terminal characteristic, 112-1 16 inverted, 246 Two-terminal characteristic problem, 193-203 computer solution of, 274-287 FORTRAN program for, 274-277, 280-287 solution algorithm for, 197-199 Two-terminal network, 111 Two-terminal subgraph, 68 reversing of, 68 Tutte, W.T., 74,309

U Unimodularness, 61-63 of incidence matrix, 63 of primitive cutset matrix, 61, 147 of primitive loop matrix, 61 of reduced incidence matrix, 63 total, 62 Uniqueness, of solutions, 101, 102

316

Index

V Van Valkenburg, M. E., 300,309 Vector, 32 minimal, 264 simple, 36 Vector space, 32,33 Vertical segment, 24 Voltage limiter, 84 Voltage source, 79,278

W Warehousing problem, 2 18-221 Whitney, H., 67, 70,74,309

2 Zigzag phenomenon, 141, see also Antizigzag rule Zoutendijk, G., vii, 309