Combinatorial Optimization (Mathematical programming study)

uX, l<-PlJ --gl,

IKI---p IJI- (p - 1)lJ n KI--- p II1, and when A, J and K are the appropriate "extreme" sets lr(A)_ J~>l q(E, ~ ) = ur(A) IKI - p" It is well-known that for every independence system (E, :g) there are matroids (E, ~:i), 1 -< i - p, such that :~ = ~:l N... 17 ~:P; we say (E, :g) is the intersection of the matroids (E, :~i). In fact let C1. . . . . Cp be the circuits of (E, :~), then (E, :g) is the intersection of the matroids (E, :~) with :~' = {A C_E: C;~ A}. Hence the following immediate corollary to Theorem 3 is of some interest:

Corollary 4. If (E, ~ ) is the intersection o[ p matroids, then q(E, ~ >- l/p. Proof. This is an immediate consequence of Theorem 3 and of the well-known fact (cf. [I]) that, for a matroid (E, ~i), the union of an independent set A E ~i and a singleton {e} contains at most one circuit. Applying Theorem 3 to some special independence systems and exploiting their special structures, we get the following results. For the standard definitions used in these results and their proofs we refer to [3, 4, 5, 6].

Theorem 5. Let Gu = (V, U) be an undirected graph and Gd = (V, D) be a directed graph (without loops or multiple edges). Let n = IvI. (a) Let 2T be the system of all matchings in Gu. I[ every connected component of Gu is a triangle or a path of length O, 1, or 2, then q( U, 2T) = 1, otherwise

q(U, ~3 =~.

D. Hausmann et aLl Greedy type algorithms

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(b) Let ~; be the system of all subsets of Hamiltonian cycles in G,. Then q(U, ~') >- 1/(2- 1/[~n]) and equality holds when G, is a complete graph. (c) Let ~: be the system of all subsets of directed Hamiltonian cycles in Ga. Then q(D, ~;) >-13. If Gd is a complete directed graph, q(D, 3~) = ~, and if Ga is a tournament, q(D, ~') equals 89or 3. (d) Let ~; be the system of all vertex packings (stable sets) in G,. Then 1/q(V, 3~) equals the maximum cardinality of a vertex packing contained in the neighborhood of some vertex v E V. In particular there is a sequence of graphs G~ =(V", U") such that IV"I= n and q(V",:~")->O for n->oo. (e) Let Gd = ( V, D) be a complete directed graph, and let ~ be the system of all the subsets of D not containing a directed cycle ("acyclic subgraphs"). Then q(D, ~') = 2In.

4. Greedy algorithm for minimum problems Since the greedy algorithm constructs a basis of E, it can also be applied to the minimum problem: minimize

c(B),

subject to

B being a basis of E.

(8)

We restrict ourselves to bases in (8), for if we would admit arbitrary independent sets as in (1), then the empty set would be an optimal solution of (8) for any (E, ~) and c. The greedy algorithm for the minimum problem (8) first orders the elements e E E such that c(el) <- c(ei+l) for any i, and then proceeds as before. Again the set G = G, will be called a greedy solution of (8). Now let crux -- max{c(e): e E E} and assume that all bases of E have the same cardinality k. Then it is easy to see that G is a greedy solution and M an optimal solution of the minimum problem (8) if[ G is a greedy solution and M an optimal solution of the maximum problem (1) for the weight function c':E-->R + defined as c'(e) = Cmax- c(e). Moreover, for any basis B of E, in particular for G and M, we have c(B) = k 9 cm~x- c'(B) Hence, Theorem 1 immediately implies the following performance guarantee of the greedy algorithm for minimum problems.

Theorem 6. Let (E, ~ ) be an independence system such that all bases of E have the same cardinality k, and let q = q(E, gk-). Let c be a weight function, Cmax= max{c(e): e E E}, G a greedy solution, and M an optimal solution of the minimum problem (8). Then c(G)<-q 9 c ( M ) + ( 1 - q). k . Cmax

(9)

and for any independence system there is a weight function such that (9) holds with equality.

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D. Hausmann et al. I Greedy type algorithms

Obviously, the bound in (9) depends not only on the independence system (E, ~-), but also on the weight function c. This shows that for any (E, ~O the worst-case behaviour of the greedy algorithm for the minimum problem (8) may be arbitrarily bad for some weight function c; cf. the corresponding examples in [3] and [6]. Two very important independence systems which satisfy the additional assumption of equicardinality of the bases of E are the systems of subsets of all travelling salesman tours (Hamiltonian cycles in a complete graph or complete digraph). Hence, T h e o r e m 5(b) and (c), and Theorem 6 immediately imply the following results:

Corollary 7. Let G be a greedy solution and M an optimal solution of the symmetric travelling salesman problem for n cities (vertices), and distances (weights on the edges) c(e)>_O, e ~ E . Then c ( G ) < - q . c ( M ) + ( 1 - q ) . n ' c m a x where q = 1 / ( 2 - 1/[89

Corollary 8. Let G be a greedy solution and M an optimal solution of the asymmetric travelling salesman problem for n cities and distances c(e), e E D. Then c(G) <- c ( M ) . 13+ (~). n . C~ax.

5. Modifications of the greedy algorithm The usual greedy algorithm, in every step, adds a single element e ~ E to the current independent set. In this section we examine modifications of the greedy heuristic which in every step add up to k elements to the current independent set where k is a fixed integer, 1 < k < n = IEI. These modifications were given earlier in Section 1. Example 1. Let Km,n be the complete bipartite graph whose vertex set R U B. Let ~ = {X C_ V: no two members in X are adjacent}. Then (V, ~ independence system where q(V, ~') -- 1/max{m, n}. Suppose [R I = m = and [B I = n = p 9 k for some integer p -> 1. Define a weight function c on c(v)=

a>O B> 0

V= is an k- 1 V by

if v E R , ifvEB.

Then R and B are the only bases of V and the basis G* obtained by either the k-starting greedy algorithm or the ( = k ) - g r e e d y algorithm must be B. Therefore c(G*) = p 9 k .ft. If a >> [(pk)2/(k - 1)] 9 fl, then p 9 k-/3 <~ (k - 1)a so M = R and c(G*) = p . k . B ~ q . c ( M ) = {~p } [ ( k - 1 ) a ] < c ( M ) . Thus, the lower bound in (3) for the usual greedy algorithm is violated. On the

D. Hausmann et all Greedy type algorithms

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other hand, the usual greedy algorithm (and the ( - k ) - g r e e d y algorithm) would find R as the greedy solution. The solution G* is so bad because exactly k elements must be taken despite the fact that it would be better to take fewer. Example 2. L e t (V, ~') be the independence system of Example 1. But suppose IR[ = m = k and IB[ = n >> k + 1. Suppose also that v0 is a particular vertex in B. Define a new weight function on V by

if u ~ R,

c(v)={i +

if u < B "-- {Uo}, forO<E
Then M = B and since 2 < k + r the solution G obtained by the usual greedy algorithm is B and so c ( G ) = ( k + e ) + ( n - 1 ) l > > 2 k = c(R). However, since 2k > (k + ~) + (k - 1) 9 1 = 2k - (1 - E), R is the solution G* obtained by the (_< k)-greedy algorithm. Thus the ( - k ) - g r e e d y solution may also be worse than (the worst) solution obtained by the usual greedy algorithm. H o w e v e r , the techniques of the proof of Theorem 1 may be modified to prove the generalization T h e o r e m 9. Let (E, ~ ) be an independence system. If c is a weight function on E, M is an optimal solution of (1), and G is any solution obtained by the

(<-k)-greedy algorithm, then c(G) >_q(E, ~-).

c(M) Proof. The (- c(e'), then i -< L The elements of E are not ordered by the (--C2>___"" >__Cn,

e~T i
~, e j E T ~§ e~ G

but

c~=cj~i<];

(10)

e~ ~ G ~c~ > cj.

Then for j = 0, 1 . . . . . n let E~ = {el, e2. . . . . ej}, Gj = G t-I Ej, and Mj = M ('1 Ej. Set c,+l=O and for j = 1,2 . . . . . n let dj=c~-cj+l. Set a ( 0 ) = 0 and for i = 1, 2 . . . . . m set a(i) to be the largest index ] such that ej E T i. If we could show n

c(G) >- ~ lr(E~)dj 9= the theorem would follow, since for any M ~ ~ we have

(11)

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.

c(M)=

q .

IMjldj j=l

--<~ q " ur(Ej)dj j=l

< ~ lr(Ej)ds j=l

where q denotes q(E, J~). Suppose a ' = a ( i - 1 ) < j < a(i) = a and suppose B is a basis of E s containing Gj. Let X = B - . Gs. If IX[ > I T i-- Gjl = t let Y be any t-subset of X and let Z=YU(Ti~.Gs). Then Z C E - - . G , [ZI=IT~ I and G U Z E ~ . Since any element of Y has a strictly larger weight than any element of T ~ --Gj, w ( Z ) > w(T') contradicting the choice of T'. H e n c e IX[ = [ B [ - IGs[ < IT' - Gj[ and

Isl < los[ + IT' -.. or[ =Ioil. From this and the fact that G~(h) is a basis of E~(h) for all h we obtain lr(Ej) ~ IGa[

for J = 0, 1. . . . . a(i).

02)

We shall show that for i = 1, 2 . . . . . m there is an index fl(i) such that setting b = / 3 ( 0 and a = c~(i) b

w( O~) >- ~'~ Ir(Ei)ds + IOa[cb+,.

(13)

j=l

Thus if a ( m ) < j < n, since G = G~m) is a basis of E s, IGIdj >-Ir(Ej)ds. From this and (12) we get

IOa(m)ICb+l:IG Ii=b+l ~ di.~j=~+

Ir(E,)d,.

Thus (13) holds for i = m, we have (ll). To prove (13) for i = 1 let H denote the usual greedy solution to (1). Since/-/s is a basis of E i in the construction, I/-/j[ - lr(Ej) for 0 --- j --- n. L e t t be the largest index j such that I/-/jf---IO~,)l = IT'I. Then

c(G,,o) = c ( T l) >- c(Ht) t

= ~ II-Ijld~+ [Htlc,+, j=l t

-->~ lr(E~)dj + [S,[c,+,. j=l

If t = n we have (13) for all i; if t < n, IHtl = [G~o)[ and putting/3(1) = t we have (13) for i = 1. Suppose now that (13) holds for i = 1, 2 . . . . . h < m ; in particular, let b be the largest index such that

D. H a u s m a n n

et a i d G r e e d y t y p e a l g o r i t h m s

129

b

(14)

c(Go) >t 2 lr(E)di + [G, Jcb+, j=l

where a denotes a(h). If b < a, then using (12)

[GalCb+, = IGa] ~

j=b+l

-> ~

dj + IGalc.+l

lr(Ej)d i +]Go[c.+1.

j=b+l

Then b could be replaced by a in (14) contradicting our selection of b. Let T denote T h+l and a' denote a(h + 1). If w(T) >_[Tlcb+l, then

c(G~,) = c(G~) + c(T) b

>--2 lr(Ei)di + Iaalcb+l + ITlcb+' 9 1=1

So setting fl(h + 1) = b we have (13) for i = h + 1. Assume then that c ( T ) < ITlcb+l and implicitly that b < n. Extend G~ using the usual greedy algorithm, i.e. set Ha = Go, and given ~ for a -< j < n, set /'/~+~ = (Hii U {eJ+~} otherwise.if ~ U {e1+1}~ ~, Let t be the largest index j such that ]Hi] <-]Goq = IGol + ]T], and let X = Ht -~ Go. Then X C E ~ Go, IxI -< ]TI and Go U X E ~:. Hence, by the construction of the (-_ IXlc,+l >-ITIcb+l> c(T). Therefore we must have a - b < t -< n, and then t

c(X)>_ 2 Ixdai + IXdc,+, j=b+l t

I1-11~. Galdi + ]Xlct+l --E j=b+l t

t

I lai- j=b+l 2 Ioolaj+ IXlc,+, -=E j=b+l t

" -j=b+l 2 lr(E/)dj -IG~l(cb+l -

Or+l)q'-]XlCt+l.

Hence c(G~,) = c(G,~) + c(T) b

>- 2 lr(Ej)di + IG~lCb+l+ c(X) j=l t

-> 2 lr(E~)dj + IG, Vct+,+ ]Xlct+l. i=1

Hence if t = n, we have (13) for all i; if t < n, we have (13) for i = h + 1. The theorem follows by induction.

IxI = ITI

and putting r

+ 1) = t

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D. Hausmann et al. I Greedy type algorithms

We have just proved that the (-
Example 3.

Consider the independence system (E, :T) where E = {1, 2 . . . . . m} form>n+2->4and = {A C_ E: A = {1,2 . . . . . n} or IA O {1, 2 . . . . .

n}l <

n}.

then q(E,50

--- l r ( E ) / u r ( E ) = n / ( m - 1).

If n = 2, then this lower bound can be achieved by the (-<2)-greedy algorithm using a constant weight function. H o w e v e r , if n = 3 this lower bound cannot be achieved by the ( - 2 ) - g r e e d y algorithm using a n y weight function c on E. On account of the s y m m e t r y in the definition of ~, we may assume that Cl~C2~C

3

and

c4>-cs>-'">-c,..

(15)

Let G be a (-<2)-greedy solution and M an optimal solution of (1). C a s e 1: G ={1,2,3}. If also M = {1, 2, 3}, then obviously c(G) _

c~

,,~ 3 ~ q t ~ , ~ = m---~-1"

(16)

Thus, we can assume that IM A {1, 2, 3}1 -< 2; hence c ( M ) <-- ca + c2 + c4 + ( m - 4)c5

Since G = {1,2,3} it follows from (15) and the definition of the (-<2)-greedy algorithm that c ( G ) = c I -]- c2 -}- c3 ~ ci -J- c2 -{- c4 --I- c5.

Thus, if c5 = 0, c ( G ) >- c ( M ) so (16) holds. If c5 > 0, then c(G) ~ cl + c2 + c4 + c5 c ( M ) - c~ + C2 q" C4 "Or( m - 4)c5 ~.

5c5 + c5

- 5c5 + (m - 4)c5 6 =m+l

3 > m- - l "

C a s e 2: [G N {1, 2, 3}1 = 2.

(as ca + c2 + c4 -> 2c3 + c4 ->-5c5 and m > 4)

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Because of (15) w e can assume that G = E ~ {3} and c4 + c5-> c3. If also IM A {1, 2, 3}[ = 2, inequality (16) follows at once, and if M = {1, 2, 3} (16) follows f r o m c ( G ) ~ CI Jr C2 Jr C4 Jr C5 ~ CI -[- C2 -~" C3 =

c(M).

References [1] J. Edmonds, "Matroids and the greedy algorithm", mimeographed notes, International Symposium on Mathematical Programming (Princeton, NJ, 1966) pp. 93-117. [2l D. Hau~mann and B. Korte, "K-Greedy algorithm for independence systems", working paper No. 7764-OR, Institut ffir 0konometrie und Operations Research, University of Bonn (Bonn, June 1977). [3] T.A. Jenkyns, "The efficacy of the "greedy" algorithm", in: Proceedings of the seventh Southeastern conference on combinatorics, graph theory, and computing pp. 341-350. [4] T.A. Jenkyns, "p-systems as intersections of matroids", in: Proceedings of the eighth Southeastern conference on combinatorics, graph theory, and computing, to appear. [5] T.A. Jenkyns, "The greedy travelling salesman's problem", Networks, to appear. [6] B. Korte and D. Hausmann, "An analysis of the greedy heuristic for independence systems", Annals of Discrete Mathematics 2 (1978) 65-74.

Mathematical Programming 12 (1980) 132-149. North-Holland Publishing Company

QUADRATIC KNAPSACK PROBLEMS* G. G A L L O lstituto M. Picone per le Applicazioni del Calcolo, CNR, Roma, Italy

P.L. HAMMER and B. SIMEONE University o[ Waterloo, Waterloo, Ont., Canada Received 17 June 1977 Revised manuscript received 10 April 1978

The quadratic knapsack (QK) model naturally arises in a variety of problems in operations research, statistics and combinatorics. Some "upper planes" for the QK problem are derived, and their different uses in a branch-and-bound scheme for solving such a problem are discussed. Some theoretical results concerning the class of all upper planes, as well as extensive computational experience, are reported.

Key words: Knapsack Problem, Quadratic Programming, Upper Planes, Branch-andBound, Computation.

I. Introduction 1.1. L e t B n be the set of all 0-1 n-vectors, and define as a quadratic 0-1

knapsack problem (QK) any of the following two problems: max s.t.

xT Qx, ax <--b, x ~ B ~,

rain s.t.

(1.1)

xr Qx, ax >- b, x ~ B ",

(1.2)

where Q is a non-negative 1 square matrix of order n, a is a positive n-vector and b a positive scalar. Since xi = x~ for x~ = 0, 1 linear terms in the objective function of (1.1) or (1.2) can be implicitly taken into account in the quadratic part. We can also a s s u m e that Q is symmetric. This model arises f r o m a variety of applications. * Presented at the IX International Symposium on Mathematical Programming, Budapest, August 1976. l Actually, the presence of negative elements on the diagonal does not affect our development. 132

G. Gallo et al./ Quadratic knapsack problems

133

Example 1. (Witzgall[14]). The technology of communication satellites has recently inspired the design of mail subsystems in which messages are transmitted electronically rather than physically. Electronic message handling stations convert physical messages into electronic ones and vice versa, and communicate with each other through satellites. Known n potential sites for the stations, the investment cost a~ for building a station in site i, and the average daily mail volume qij observed between i and j, the problem of selecting a set S C_{1..... N} of locations such that the global traffic ~ . j e s q~j is maximized and a budget constraint ~,ies ai <-b is met is obviously of the form (1.1). Similar problems have been investigated regarding the location of railway stations (Land [8]), freight handling terminals (Rhys [11]) and airports. Example 2. In hydrological studies, the observations made by different pluviometers in a same region are usually seen to be correlated. Thus it is desirable to reduce the number of puviometers by choosing only a few of them, with minimal redundancy, and an acceptable loss of information. If qij, iS j, is the absolute value of the covariance between the rainfall in stations i and ], estimated from a record of past observations, qii -- 0, ai is the variance of rainfall in i, and b is a fixed fraction of the total variance ~,~ai, the problem can be formulated as (1.2). A similar model arises in applications to portfolio selections (Laughunn [9]), when one wishes to select a subset of n possible investment proposals such that some fixed minimum return is guaranteed and the risk is minimized. The sum of the absolute values of the covariances of the (random) returns associated to the proposals can be taken as a reasonable measure of risk. Example 3. The problem of determining whether a given graph possesses a clique of order k (k is a fixed integer) is a special case of (1.1), where Q is the node-node adjacency matrix of the graph, ai = 1 for all i and b -- k. The clique does exist if and only if the optimum value of (1.1) is k(k - 1). It is well-known (Karp [7]) that this clique problem is NP-complete. More generally, the QK problem can be formulated in the following graph-theoretic terms. Given a graph G, let a "benefit" be associated with each edge, and a "cost" with each node of G. Define the "benefit" of a subgraph H to be the sum of the benefits of all edges in H, and the "cost" of H to be the sum of the costs of all nodes in H. (1.1) is then equivalent to the problem of finding a subset S of nodes, such that the subgraph induced by S has a cost not exceeding some fixed level, and yields a maximum benefit. 1.2. The present paper is mainly concerned with the formulation (1.1) of QK. Actually, the algorithms we propose can be converted, in a straightforward way, for handling problem (1.2). Any lagrangian relaxation of (1.1) with respect to the knapsack constraint is amenable to the maximization over B" of a quadratic function whose diagonal

G. Gallo et al. I Quadratic knapsack problems

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terms are negative and whose off-diagonal terms are non-negative. Rys [11] showed that the last problem is equivalent to a network flow one and Balinski [1] proposed a labelling procedure for solving it. 1.3. The algorithms developed in this paper are based on bounds on the optimum value of (1.1), obtained by considering linear relaxations in which the quadratic objective [(x) is replaced by an "upper plane") i.e., a linear function g(x) which dominates f(x) in all feas~le points. This approach is motivated by the easiness with which even large linear knapsack problems can be solved (Salkin and de Kluyver[19]). The use of upper planes is not new in the literature (Taha [13]) and goes back to at least as early as to the works of Beale [3] and Balinski [1] on certain nonlinear transportation problems. Upper planes have usually been considered in connection with the outer-linearization of a concave function; also, Cabot and Francis [4] suggested a simple method for deriving upper planes for non-convex continuous quadratic problems. (As a matter of fact, the upper planes constructed in Section 2 of this paper turn out to be closely related to those of Cabot and Francis.) The focus of this work is on the relative efficiency of different upper planes and on the different ways in which they can be exploited in branch-and-bound techniques for solving (1.1). Extensive computational experience is presented. Finally, in the last section some theoretical properties concerning the family of all upper planes are investigated.

2. Upper planes for the quadratic knapsack problem 2.1. Let us denote by X =- {x E Bn: ax <- b} the feasible set of (1.1). Without loss of generality, we can assume ~7~ ai a~ > b > a~ -> a2->" 9 9>-- an > 0. An upper plane (UP) for the function f ( x ) = xTQx in X is any linear function g(x) such that g(x) >_[(.x) for all x ~ X. Given an UP g(x) for f(x) in X, the corresponding linear relaxation of QK is the (linear) knapsack problem max{g(x): x E X}.

(2.1)

From the solution of (2.1) both an upper and a lower bound on the optimum value of (1.1) can be derived. Indeed, if x* and ~ are optimal solutions of (1.1) and (2.1) respectively, one has f($) <- f(x*) <- g(fr 2.2. Given the upper plane g(x), a cheaper way of getting bounds consists in solving the continuous knapsack problem max{g(x): x ~ 3(), where f ( - { x E R n :

ax<-B, O<_x<_e}, with e =(1, 1..... 1).

2Some authors use the term "linear overestimator'.

(2.2)

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135

Such a problem admits an optimal solution whose components are all zeroes or one, with the possible exception of a single component. Setting this variable down to zero, a binary feasible vector x ~ is obtained and f ( x ~ is a lower bound for the quadratic optimum; an upper bound is given by the optimum value of (2.2). 2.3. The algorithm which will be described in the next section is an enumerative one, in which the exploration of each node can be viewed as a two-stage process. The r61e of the first stage is the generation (through the exact or approximate solution of a number of linear knapsack problems) of the coefficients of an upper plane, while the r61e of the second stage is the determination of upper and lower bounds on the maximum of f ( x ) in X (through the solution of the linear relaxation corresponding to the above upper plane). 2.4. A simple way to derive upper planes is the following. If vi is an upper bound of the set {qTx: X ~ X}, where qi is the jth column of Q, the function ~7=1 vjxj is obviously an U P f o r / ( x ) = ~,~=1 (qTx)xi in X. This turns out to be true even when vj is an upper bound for the set {qTx: X E X, xj--1}, since (qTx)xi <-- VgCj when xi = 1 and ( q ~ x ) x i = vixi = 0 when x~ = 0. Four possible choices for vi are given, under the hypothesis that qii -> 0 for all i,j:

v 2 = ~,~h)qi~,

(2)

where ~!h~ means that the sum is restricted to the h largest elements of the jth column of Q, and h is the maximum number of ones in a feasible solution (since the a~'s are non-increasing, h is the largest index k for which an-k+1 + " " + an <-b), v~ = max{qTx: x E ,~},

(3)

(of course one can take the integer part of the r.h.s, if Q is integer), v~ = max{qTx: x E X}.

(4)

The order relationship between the four upper bounds is shown in Fig. 1, where a continuous arrow points from a smaller to a larger upper bound, while the dotted arrow means that in most observed cases (but not always) v] was found to be smaller than v~.t3) On the other hand, the computational effort needed for evaluating the vfs increases from (1) to (4). The upper bound (4) is the best possible one, but it is also by far the most expensive. Extensive numerical experimentation has been 3A remarkable exception arises when all the maximalfeasible vectors have the same number of ones. In this case v~= v~-> v~.

G. Gatlo et al./ Quadratic knapsack problems

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Fig. 1. performed (see Section 4) in order to find an upper plane which corresponds to a good trade-off between computational effort and tightness to the quadratic function. 2.5. A given lower bound on the optimum value of (1.1) can often be improved, at low cost, through a sequence of the following elementary operations (suggested for linear knapsack problems by Petersen [10]): F i l l - u p , which transforms a given feasible point x, having xi = 0, to a point of the form x ' = x + uj (uj is the jth unit vector). Then one has f ( x ' ) _ f ( x ) , the inequality being strict unless qj is the zero vector. E x c h a n g e , which transforms a point x such that xi = I and x s = 0 for some i, j to the point x' = x - ul + uj, so that x~ -- 0 and x~ = 1. The exchange is performed only if x' is still feasible and (using the terminology in [5]) the s e c o n d d e r i v a t i v e

(

Aii(x) = [ xl . . . . . 1 . . . . . . . . .

, x,

)

i

-,(x, ..... o ..... 1 ..... x.) is negative. Since Aii(x) = A~(x) - As(x) + 2qis(xi - x~),

where Ak(X)=

,(

k

xl . . . . .

'1 . . . . . x , ),(--

k

'

xl . . . . . 0 . . . . . x ,

) = qu + 2

h~k

qh~xh

is the f i r s t derivative, only the first derivatives need to be kept in storage; they can be updated after each elementary operation by simple formulae. There are two good reasons for trying to improve the lower bounds. As we shall see later, the above discussed linear relaxations yield very good l o w e r bounds. Thus, the solution of the linear relaxation, followed by the improving procedure sketched above, gives rise to an economic and effective h e u r i s t i c a l g o r i t h m , which often leads to the true optimum. If, on the other hand, one wishes to use linear relaxations in a branch-and-bound scheme for finding an optimal solution, early knowledge of tight lower bounds enhances the fathoming power of the algorithm.

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137

3. Use of upper planes in branch-and-bound algorithms 3.1. Upper planes can be exploited as basic tools in deriving enumerative algorithms for QK. For any subproblem obtained from (1.1) by restricting certain variables to 0 or 1, it is possible to derive bounds through the solution of a discrete or continuous linear relaxation. Moreover, upper planes play an interesting r61e in other basic aspects of enumerative algorithms, such as forcing of variables and branching. Up and down penalties for linear relaxations can also be obtained with a modest cost. Finally, the amount of work required in order to generate the linear relaxations can be considerably reduced by means of recursive formulae. Throughout this section, we shall assume, for the sake of simplicity, that all the data Q, a, b are integer; we shall denote by U, Z and F the index sets of the variables which, in the current subproblem (P), are fixed to 1, 0 or are free, respectively, so that the feasible set in (P) is

S-{x:

xEX;xj

= I ViE U:xi=O Vj~Z};

Vo + ~,ieF V~Xi will stand for the current UP, and ~,]~F a~xj < G for the current knapsack constraint, while (p0) and (P~) will represent two subproblems, with feasible sets S ~ and S~ respectively, obtained from (P) by adding the further constraint xr = 0 or xr = I. 3.2. Bounding. In each subproblem (P), the restriction of the objective function f to S is a non-negative quadratic function in the free variables, for which we can construct an upper plane VO+~,iEF ViXj, according to one of the rules (1) ..... (4) of Section 2.4. It is important to recognize that, in all four cases, the law which associates to any subproblem (P) the corresponding upper plane g is adaptive: in other words, if g' and g" are the upper planes corresponding to (p0) and (P~), generated according to the same rule as for g, one has g'(x) <-g(x) Vx E S ~ and g"(x) <- g(x) Vx E S~. Let us prove it, for illustration purposes, in the case that g(x) is the UP v4x. Let us also restrict our attention to the upper plane g'(x) = v"x for (P~). Dropping for simplicity the index "4", we have: Vo = E

i,]EU

qii,

Vi=2 iEU E qii+max~Eqi~xi: Eaixi<--b, xi=O,l~,.I ~.iEF iEF

v'o = i.~ff,avqij + q=, v~=2~

i~U

qii + 2q,j + max{i ~ ~F-{r} qijxi: ~,

iEF-{r}

aixi-----/~- a,, xi = 0, 1}.

Hence we have vo= v ~ - q . ,

vi >- q,i + (vj - 2@0 = v; - q,j, V] E F - {r},

G. Gallo et al./ Quadratic knapsack problems

138

so that, for any x ~ S~,

g(x) = Vo+ Vr+ E

i~F--{r}

IE

{r}

/ \ ViXj> (V~-- qrr)+ [qrr+ E qirXi) / \ iEF-{r} j~F-{r}

jEF-{r)

3.3. Branching. Let x ~ be an optimal solution for the linear relaxation of the current subproblem, and let F~---{j~ F: x ~ 1}. A reasonable criterion for selecting the branching variable x, is to choose r such that

v-r' = ar

max

v/

lEFt ai'

the branch x, = 0 being explored first. 3.4. Recursions. If V~+~,j~FV~Xj is the U P for the current node, generated according to rule (1) of Section 2.4, the corresponding UPs for the two successors can be generated through simple recursions. Actually, a direct computation shows that:

Proposition 3.1. The UPs of type (1) for (p0) and (P~) are given by v~ + j~_~,~ ( v ~ - q.)x~ and by

respectively. Slightly more complicated recursions hold for the UPs (2) and (3). Even if recursions are not used, it is important to notice that the most expensive part of the work required for generating the U P (3), i.e., sorting the elements of each column j according to non-increasing ratios qq,/a~, need not be repeated, once it has been done for the initial one. A similar observation can be done for the UP

(2). 3.5. Forcing. At each node, one is really interested only in those feasible points x in which the quadratic objective function, and hence, afortiori, the current UP, is strictly greater than the current lower bound z. The constraint v0+ ~jep vjx~>_ _z+ l, combined with the knapsack constraint ~,jeF a~xj <- b, may be very effective in forcing variables to the value 0 or 1. To this end, the constraint pairing techniques described in Hammer et al. [6] can be successfully exploited. Such techniques also allow for detecting whether the system composed by the two above inequalities is inconsistent.

G. Gallo et al.l Quadratic knapsack problems

139

3.6. Penalties. Let u, u ~ and u~ denote upper bounds for the subproblems (P), (p0), (p~) respectively, computed according to one of the methods of Section 2.4. Down and up penalties, i.e., underestimates of the decreases u - u ~ and u - u~ can be exploited, in the usual way, for fathoming (P) by bound, for forcing variables o r - a s an alternative to the criterion in Section 3.3- for selecting the branching variable. The following proposition provides an inexpensive procedure for evaluating down and up penalties when the upper bounds are obtained by solving continuous linear relaxations. Such penalties are valid for an arbitrary upper plane, provided that it is adaptive (according to the definition in Section 3.2). Let Vo+ ~,i~F V~X~be such an UP for (P). Assuming the free variables numbered according to non-increasing ratios vr/aj, let xt be the last positive variable in the optimal solution $ of the corresponding continuous linear relaxation of (P). Proposition 3.2. (a)

If ~,

= 1,

(\ Vr- a~vt at) is a down penalty for Xr. (b) II ~, = O, a,-

v,)

is an up penalty for x, (c) If 0 < 2r < 1 (i.e., ! = r),

,~,(v, -~l+lU,] V,+l_~

and

(1 -,~,)(~

at - v , )

are down and up penalties for x , respectively. Let us prove, for example, (a). If ~, = l, since the upper plane is adaptive we have

U~176

i~F ~ a~xj <-b;O<-xi <- l' Vj E

(3.1)

On the other hand,

U=Vr+max~vo+ t.

~,

jEF-~r}

v~xF ~

jEF-{r}

ajxj<-b-ar;O<-xj<-l,VjEF-{r}}

>-V, _v~ at ar+ max{vo +/eF~ } ViXF i~-,,, ~ a j x i < - b ; O < - x i < - l ' V j E F - { r } '}

G. Gallo et al./ Quadratic knapsack problems

140

that is,

u>-v,-Vla,+max{vo+~VFj:~,aFj<-G;O<-xi<-l, Vj~F;xr=O}. at ~Ep I~F Comparing the last inequality with (3.1), one gets

u-u~

at

which proves (a). Statements (b) and (c) can be proved in a similar way. 3.7. We now give a general description of a class of branch-and-bound methods for QK, involving the procedures discussed in this section. Two algorithms in the class may differ in one or more of the following aspects: - - t h e type of upper plane ((1), (2), (3) or (4) of Section 2.2) generated at every node during the first stage (see Section 2.1). the kind (discrete or continuous) of linear relaxation solved at every node during the second stage. the presence or absence of routines for constraint pairing and for evaluating and using penalties. Each algorithm performs a standard search over a binary tree, whose nodes correspond to subproblems obtained by restricting some variables to the values 0 or 1. The two successors of each node are generated by fixing some free variable to 0 or to 1, respectively. In exploring the tree, a LIFO priority rule is adopted. At each node the following steps are performed:

Step 1: Step 2:

A linear relaxation for the corresponding subproblem is generated. An attempt is made in order either to detect infeasibility or to force a single variable by the constraint pairing techniques mentioned in Section 3.4. If the problem is feasible and no variable can be forced, Step 3: The relaxed subproblem is solved and the bounds on the quadratic optimum are updated. Step 4: Every time the lower bound is improved, use is made of the routine described in Section 2.5 in an attempt to further improving it. Step 5: If the subproblem cannot be fathomed by bound or by optimality, Step 6: Up and down penalties for all the free variables are computed, by means of which the algorithm again tries either to fathom the subproblem by bound or to force as many variables as possible. In case of failure, Step 7: A branching is made on the variable with the highest up or down penalty. Step 8: (Only if Steps 6 and 7 are not executed) A branching is made on the variable selected according to the criterion of Section 3. Step 9: Whenever a subproblem is fathomed (by infeasibility, bound or

G. Gallo et al.I Quadratic knapsack problems

141

optimality), backtracking to the predecessor occurs and the other branch emanating from it is explored, unless this has already been done. Steps 2, 6, 7 are optional.

4. Computational experience 4.1. Numerical experimentation has been performed on ten different variants of the general branch-and-bound method sketched at the end of the last section. To this end, a FORTRAN code for this class of algorithms has been implemented on an IBM 370/168. For reference, each variant is identified by a label, in which the first character is B or C, according to whether discrete or continuous linear relaxations are solved during the second stage at each node; the second character is 1, 2, 3 or 4, according to the type of upper plane generated during the first stage; one or more optional characters may follow, with the following meaning: F a routine for forcing variables, based on constraint pairing techniques, is included. Fk as above, but only k surrogate constraints are generated rather than n + 2 as suggested in [6]. P a routine for evaluating and using penalties is included. 4.2. The numerical experiments were performed on randomly generated test problems, ranging from 10 to 70 variables, in which the elements of Q were integers uniformly distributed between 0 and 100, the knapsack coefficients aj were integers uniformly distributed between 1 and 50, and the b's were integers uniformly distributed between 1 and E~=~aj. Unless otherwise specified, Q was taken full (100% dense). For each problem size n -- 10, 20 ..... ten test problems were run. 4.3. Let UB and LB denote the upper and lower bounds for the initial subproblem, computed as in 2.1 and 2.2, and let OPT denote the quadratic optimum. Table 1 gives, for the variants B1, B2, B3, B4, C3, the average "Av", the standard deviation "Sd" and the maximum "Max" (estimated, for each size n, from a sample of ten test problems) of (UB-LB)/LB% and of (OPT-

LB)/LB%. The following remarks are in order: - - T h e upper bounds given by B4 and B3 are remarkably better than those provided by B2 and B 1. - - T h e lower bounds are always much closer to the quadratic optimum than the upper bounds. - - T h e relative error of the bounds with respect to the optimum appears to remain of the same order of magnitude as the size n of the problem increases.

G. Gallo et all Quadratic knapsack problems

142 Table I

UB - LB.% LB

OPT - LB% LB

Version

n

Av

Sd

Max

BI

10 20 30

75.42 166.02 74.01

82.77 229.03 92.09

293.48 835.05 307.47

B2

10 20 30 40

25.73 39.61 31.02 32.96

14.30 22.97 28.44 22.30

B3

l0 20 30 40 50

17.89 12.62 10.50 13.46 10.88

B4

10 20 30 40

C3

10 20 30 40 50

Av

Sd

Max 9

8.52 1.03 0.63

16.16 2.45 1.69

51.55 8.30 5.68

42.24 76.46 97.39 69.74

3.80 3.82 0.41 1.76

5.73 8.28 0.86 2.38

15.09 28.43 2.69 7.24

19.80 6.08 4.68 5.31 5.77

73.31 20.32 14.73 20.54 18.91

6.40 0.56 0.15 0.20 0.08

17.92 0.70 0.29 0.37 0.10

50.28 1.24 0.79 1.14 0.22

11.40 10.83 9.66 10.87

6.13 5.60 4.17 6.66

21.55 17.87 14.24 16.68

1.43 0.55 0.14 0.49

3.16 0.70 0.28 0.35

10.01 1.76 0.78 0.75

35.47 21.46 17.86 16.55 16.49

31.79 14.45 11.98 6.68 7.46

111.33 52.05 49.27 26.89 30.21

10,34 5.51 3.90 1,31 2.75

15.22 8.30 6.32 1.20 3.33

51.56 26.83 22.45 4.43 10.74

The a c c u r a c y of lower bounds can be further improved, as Table 2 shows, when use is m a d e of the routine sketched in Section 2.5. The column " S c o r e " shows in how m a n y cases out of ten the initial lower bounds was found to be

equal to the true quadratic optimum. Table 2

OPT - LB% LB Version

n

Av

Sd

Max

Score

B3

10 20 30 40 50

4.39 0.23 0.08 0.19 0.04

12.64 0.46 0.24 0.36 0.09

42.29 ! .24 0.79 1.14 0.22

8 8 9 7 8

C3

10 20 30 40 50 60

3.17 0.78 0.29 0.47 0.90 0.34

6.35 2.00 0.45 0.47 1.82 0.87

16.02 6.70 1.07 1.38 4.97 2.94

8 8 7 3 8 6

143

G. Gallo et al. ] Quadratic knapsack problems

Table 3 n

10

20

30

40

0.09 0.08 0.04 0.09 0.056 0.036 0.033

1.65 0.85 0.40 0.49 0.23 0.19 0.17

0.04 0.042

0.21 0.25

21.28 8.78 3.93 5.89 2.56 1.54 1.17 1.53 1.13 1.23

37.34 43.99 14.45 9.46 7.88 9.31 8.24 9.18

50

60

70

78.83 27.93 29.22

68.92 78.36

318.714

Version B 1F B2 B2F

B3 B3F

C3 C3F C3F3 C3P C3FIP

27.78 30.30

Notice that, with this improvement, the lower bounds given by C3 become comparable with those given by B3. In Tables 3 and 4, the performance of the above mentioned ten variants 4 of the algorithm is compared in terms of average running time per problem (in CPU seconds on the IBM 370/168) and in terms of average number of linear relaxations generated (upper figure) and solved (lower figure) during the branch-andbound search. The averages have been estimated, as usual, from a sample of ten test problems for each size n. It should be remarked that, since forcing a variable Xr to the value a (a = 0 or 1) is always followed by a forced branching along the opposite direction xr = 1 - a, (which implies the generation of a new relaxation and thus delays a possible fathoming by bound), the number of linear relaxations generated is usually higher in those variants which include forcing routines. The following remarks can be made: The algorithms' performance is heavily conditioned by the particular type of U P chosen. - - T h e C3 variants unquestionably outperform the other ones. None of the variants C3, C 3 F and C3F3 seems to be definitely dominant over the other ones in terms of running time. - - T h e forcing routine tends to increase the number of linear relaxations generated and to decrease the number of those solved. While it is very effective in B type variants (with savings of 50% and more) it loses its effectiveness in C type variants, where the times for generating a linear relaxation are roughly comparable with those needed for solving it. - - U s e of penalties sensibly reduces the number of nodes explored, but obviously increases the amount of work per node. In the range of problem 4 Because of its expensiveness, variant B4 was only used to solve the initial problem.

144

G. Gallo et al. / Quadratic knapsack problems Table 4 n

10

20

30

40

50

60

70

B 1F

65.4 12.7

457.9 77.7

B2

18.8 18.8

63.4 62.9

227.4 227

B2F

29.4 7

108.2 17.5

453.4 66.7

B3

23.2 20.4

38.6 32.3

147.6 137

533.2 526.8

B3F

10.2 2.8

45.6 6.4

215.2 34.5

991.2 126. l

3235.4 361.8

C3

12.12 12.12

58 53.8

259 251.8

1138.6 1130.6

2094.4 2079.8

3476.4 3449.1

C3F

10.44 2.82

50.6 7.6

231 38.9

1257.7 181.4

3329 321.3

5556.8 530.9

239.5 56.6

1121.5 284.4

Version 2868 426

C3F3

2480 322

C3P

7.4 7.3

36 36

110.2 110.2

587.4 587.4

1397 1396.9

C3FIP

8.2 6.4

51.6 29. l

136.6 87.9

785.8 453.4

2094.9 885.5

15611 1787.8

sizes observed, there seems to be no particular advantage in using them. The variant in which they are used together with constraint pairing is not recommended. Tables 5 and 6 show, for some variants, the effect of decreasing the density of Table 5

UBLBLB %(A v )

OPT B LB %(A v )

Size Density

10

20

30

40

50

l0

20

30

40

50

100% 50%

17.89 27.88

12.62 39.63

10.50 27.92

13.46 37.50

10.88

6.40 0

0.23 1.13

0,08 1.53

0.20 1.05

0.04

25% 5%

38.27 9.63

52.12 21.39

44.28 34.77

31.44 39.90

0 0

1.29 2.92

5,54 4,34

0.22 4.87

35.47 38.63 70.60 16.55

21,46 43.62 53,62 23.36

17.86 28.37 40.53 36.44

16.55 39.57 53.64 36.49

10.34 1.12 7.95

5.51 2.02 0.58 2.05

3.90 1.86 3.18 4.53

1.31 1.65 3.72 3.01

Version B3

28.97

3.43

Version C3 100% 50% 25% 5%

16.49

28.06

2.75

2.49

G. Gallo et al./ Quadratic knapsack problems

145

Table 6 Version B3F Size: Density: 100% 50% 25% 5%

10

20

30

40

0.056 0.047 0.040 0.026

0.23 0.65 0.54 0.11

2.56 6.38 4.53 0.70

14.45 88.50 74.55 5.40

10

20

30

40

0.036 0.039 0.037 0.021

0.20 0.55 0.52 0.12

1.54 6.81 3.99 0.79

9.47 71.71 90.42 6.61

10

20

30

40

0.033 0.034 0.029 0.018

0.18 0.40 0.30 0.088

1.17 3.93 2.04 0.45

7.88 43.02 50.92 3.28

Version C3 Size: Density: 100% 50% 25% 5% Version C3F Size: Density: 100% 50% 25% 5%

Q on the tightness of the bounds (Table 5) and on the average running time per problem (Table 6). All variants are seen to behave better when the matrix is very dense (100%) or very sparse (5%). Fortunately this is actually the case in most applications.

5. Some theoretical properties of upper planes 5.1. Let S be any subset of R n and f an arbitrary real function whose domain contains S and which has a maximum in S. For convenience, we shall often denote a point V - (v0, v~. . . . . vn) in R n+~ as (v0, v). Define S*~{VERn+I:

v0+vx~>f(x),xES}

by convention, 4'* = R "+~. The following properties hold:

(a) (b) (c) (d) (e)

S D_T ~ S * C_ T*; (S U T)* = S* n T*; (S n T)* _DS* U T*; S* is non empty, convex and unbounded;

If S is finite, S* is a convex polyhedron;

G. Gallo et al. I Quadratic k n a p s a c k problems

146

(f) If f is c o n v e x and S is a c o n v e x polytope (i.e., a bounded convex polyhedron), then S* = E ( S ) * , where E ( S ) is the set of the vertices of S; (g) If [ is c o n v e x and S is a c o n v e x polytope, S* is a c o n v e x polyhedron. The proof of properties (a) through (e) is straightforward s. (g) follows at once from (e) and (f). L e t us p r o v e (f). By property (a), E ( S ) * 3_ S*. On the other hand, let V =- (vo, v) E E(S)*. Any x E S is a c o n v e x combination A~Zl + -.. + ;t,z, of the vertices of S. H e n c e vo + vx = vo

At + v

Xiz~ =

"=

,li(vo + vz~) >i=1

~J(z~) i=l

Thus V E S* and E ( S ) * = S*. 5.2. For simplicity, we shall assume n o w S to be a c o m p a c t set. For any V E S*, maxxas(v0+ vx) does exist and is an upper bound for z = m a x x ~ s f ( X ) . On the other hand, there is at least one I) E S* for which such an u p p e r bound actually equals z (it suffices to take I7"= (z, 0 . . . . . 0)9. H e n c e we have

Proposition 5.1. min max(v0 + vx) = max f ( x ) . VES* xES

x~S

This equality suggests that the m a x i m u m of f in S could be found, in principle, by minimizing o v e r S* the convex function h ( V ) = maxx~s(V0 + vx). Notice that, if h ( V ) = vo+v$, then [1 " ~] is a subgradient of h in V, since, for any V ' E S * , we have h ( V ' ) > v ~ + v ' ~ = h ( V ) + [ 1 i ~ ] ( V ' - V).

Corollary 5.2. I f h(17) = minv~s, h( V) and [ ( s = maxx~s/(x), then Vo + ~x attains its m a x i m u m over S in s Proof. for all x ~ S, ~50+ v.f -< h ( f ' ) = f(.f) ~ z5o + ~ . 5.3. Remark. It is worth noting that, if we restrict ourselves to homogeneous U P s vx, it is still true that m i n ( m a x vx: v ~ R", vx >- f ( x ) u k xES

~ S } >- m a x f ( x ) , J

(5.1)

xES

but equality does not h a v e to hold, as the following example shows: 5 Properties (a), (b), (c) are satisfied, more generally, by any Galois connexion between the subsets of R" and the subsets of R"+L

G. Gallo et al./ Quadratic knapsack problems

147

Example. f ( x ) = 18x~x2 + 2xtx4 + 2x~x5 + 2x2x3 + 2x2x6 + lOx3x4 + 2x3x6 + 2x4x5 + 10xsx6,

S ~ {x E B*: 5x~ + 4x2 + 4x3 + 2x4 + 3x5 + 2x6 -< 10}. The maximum of [ ( x ) in S is 18. Equality in (5.1) would imply the existence of a vector v such that 18 <-- v~ + v2 --< 18, 10 --< v3 + v4 -- 18, 10_< vs+ v6-< 18, 6 <-- vt + v4+ v5 <-- 18,

6--< v2+ v3+ v6 -< 18, since the points (1, 1, 0, 0, 0, 0), (0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 1), (1, 0, 0, 1, 1, 0), (0, 1, 1, 0, 0, 1) belong to S. But the above system of inequalities is clearly inconsistent.

5.4. Let us now come back to the particular case in which [ ( x ) = - - x r Q x and S-= X = {x E Bn: a x <-b}. Since X is finite, X* is a convex polyhedron by the above property (e). An explicit description of X* as intersection of half spaces is in practice very difficult to obtain, because of the usually large number of points in X. However, property (a) above suggests the possibility of getting more tractable subsets of X* from suitable extensions of X. Let us consider, for instance, the extension : ( ~- {x E Rn: x >- O, ax <- b} D X. Define p as the vector whose jth component is (b/aj)qj~.

Theorem 5.3. It' Q is positive semidellnite, p x is an U P f o r x T Q x in X. Moreover, h(0, p)=

min h(vo, v). (v0, v )EX*

Proof. 2* = E(X)* =

{

O,

u~. . . . . ~ un

a

}

= ( vo, v ): vo > O, vj + _~ vo ---~qjj b for j = 1. . . . . n . From the last expression of X* it is apparent that the point (0, p) belongs to

148

G. Gallo et al./ Quadratic knapsack problems

.~*. On the other hand, for all (v0, v) ~ .~* and x E ~, one has v0+ ~=~ ~ vpcj-> v0+ j=l b q, _a_L b Vo xj = vo l - -

a/x/

n

n

]~l

j=l

This proves the second half of the theorem. The upper plane p x has been tested on randomly generated test problems ranging from 10 to 80 variables. The results, except for some very tight problems (i.e., with b -< 0.2 ~7=J a~), showed a poor behaviour, due perhaps to the fact that in most observed cases there were a few p f s taking large values, which have sharply pushed up the optimum value of the linear relaxation, in spite of the fact that the vast majority of the p's were low. A slight improvement was obtained by replacing the above considered X with )(, - {x ~ R" : x - O, a x <-- b, xr <- 1}, where p, = max~_~i<, pj. It is hoped that tighter extensions of X could lead to significant improvements.

Acknowledgement Acknowledgements are expressed for the partial support given by the Istituoto per le Applicazioni del Calcolo (Rome) and the Centro Nazionale Universitario di Calcolo Elettronico (Pisa) of the Consiglio Nazionale delle Recherche, as well as the National Research Council of Canada (Grant A8552) and Nato's System Science Exchange Grant.

References [1] M.L. Balinski, "Fixed cost transportation problems", Naval Research Logistics Quarterly 11 (1961) 41-54. [2] M.L. Balinski, "On a selection problem", Management Science 17 (3) (1970) 230-231. [3] E.M.L. Beale, "An algorithm for solving the transportation problem when the shipping cost over each route is convex", Naval Research Logistics Quarterly 6 (1959) 43-56. [4] V.A. Cabot and R. Francis, "Solving certain non-convex quadratic minimization problems by ranking the extreme points", Operations Research 18 (1970) 82-86. [5] P.L. Hammer and P. Hansen "Quadratic 0-1 programming", discussion paper No. 7129, CORE. (Heverlee, 1972). [6] P.L. Hammer, M. Padberg and U.N. Peled, "Constraint pairing in integer programming", INFOR. Canadian Journal of Operational Research and Information Processing 13 (1975) 68-81. [7] R.M. Karp, "Reducibility among combinatorial problems", technical report 3, Computer Science University of California (Berkeley, CA, 1972). [8] A. Land, personal communication (1975). [9] D.J. Laughhunn, "Quadratic binary programming with applications to capital budgeting problems", Operations Research 18 (1970) 454-461. [10] C.C. Petersen, "A capital budgeting heuristic algorithm using exchange operations", AIIE Transactions 6 (1974) 143-150.

G. Gallo et al./ Quadratic knapsack problems

149

[1 I] J. Rhys, "A selection problem of shared fixed costs and network flows", Management Science 17 (3) (1970) 200-207. [12] H. Salkin and C.A. de Kluyver, "The knapsack problem: a survey", Naval Research Logistics Quarterly 22 (1) (1975) 127-144. [13] H.A. Taha, "Concave minimization over a convex polyhedron", Naval Research Logistics Quarterly 20 (1973) 533-548. [14] C. Witzgall, "Mathematical methods of site selection for Electronic Message Systems (EMS)", NBS internal report (1975).

Mathematical Programming 12 (1980) 150-162. North-Holland Publishing Company

FRACTIONAL VERTICES, CUTS AND FACETS OF THE SIMPLE PLANT LOCATION PROBLEM Monique GUIGNARD University of Pennsylvania, Philadelphia, PA, U.S.A.

Received 22 May 1978 Revised manuscript received 3 January 1979

This paper investigates the structure of the Integer Programming Polytope of an uncapacitated (simple) plant location problem. One can describe families of fractional vertices and derive from them valid inequalities for the integer problem. Some of these will actually be shown to be facets of the integer polytope. Also some families of SPLP with large duality gaps will be described, together with facets which bridge these gaps. Much of the motivation stems from algorithmic work in which the exploitation of "good" cutting planes within a direct dual algorithm have been shown to be of crucial importance. Key words: Plant Location Problem, Cutting Planes, Facets, Convex Polytopes, Duality Gap.

O. Introduction This paper investigates the structure of the L P polytope of an uncapacitated location problem: SPLP(m, n):

min~ ~

x,y i~l jEJ

cijxi~+ ~,

f/y/,

iEl

VjEI: ~

x/~

ViEI, VjEJ:

X~j-

=l,

iEl

yi-<0, -0,

Xi/

y/-0,

integer,

where I is a set of m possible plant locations, J a set of n destination points, xij the ratio of the requirement of j E J satisfied from location i E / , and y/, a 0-1 variable, is 1 if and only if location i E I is open. c~j is a normalized cost of shipment from i to j, and f/ is a fixed charge associated with opening plant (or location) i E L We assume n --> m >- 3. We will first construct families of fractional vertices of the L P polytope, and generate from them valid cuts. We will then prove that some of these cuts are facets. 150

Monique Guignard/ Plant location problems

151

In a second part, we will study families of S P L P with large duality gaps, and show that for these one specific facet bridges the gap. Fractional vertices have been characterized in [2, Theorem 8], where certain cuts are proposed without proof. In this paper we construct similar inequalities and derive from them cuts which are shown to be facets of the integer polyhedron. We have shown elsewhere, [4], how such cuts can be (and indeed must be) exploited via their associated dual variables within a direct dual algorithm (for the more difficult capacitated plant location problem and for pure integer cuts) and expect that a similar approach may be even more effective when suitable facets are constructed and utilized during the calculations.

1. Some fractional vertices of SPLP (m, n) Define yz = 1 - y; and s~j = 1 - x0 - Yi, Yi ~ L Vj E J. SPLP(m, n) can be rewritten as a set partitioning problem: m +minx

X ci~xij-XfYi

WjSJ: ~ iEl

x0

=

ViELVj~J:

x~i

+Yi + sij = 1 (*it)

x,~ iEl jEJ

iEl

l

(~i)

yi, so->O y~ integer. LP(m, n) will denote the LP relaxation of SPLP(m, n). Let o- be a surjection (i.e. an onto function) from J into I p-a - I ~ ~ where li~l=p>O, m>p+2, and let M i = { j E J l c r ( j ) = i } , i E I p , 1Vli=Y~.Mi. Let Jq _CJ be such that the restriction of cr to Jq, cr]Jq, is still a surjection into Ip. Let J, = J ~ Jq, where q = I . ~ l>- O. We will need the following results in the generation of the valid inequalities. Lemma 1.1. Vi # i', i, i' E Ip, M~ f3 Jq C JQlv n Jq.

Proof. Let i, i' ~ Ip. i" ~ Ip (1) V i ' # i, M / f 3 J q # / ~ , , N Jq: since or is a function (o'(i) is unique for a given i)

Yi" # i, i' (i" exists as lip[ > 2) i" E Ip M,. n ]~ c ]~ ~ ( M, UM/,) If M~ n Jq = M,-,n Jq, then M,., n Jq = 0 which contradicts the fact that o'lJq is a surjection.

Monique Guignard/Plant location problems

152

(2) V i ~ i ' , M ~ A J q C MI~,NJq

V j E M ~ A J q , Vi' ~ i,

~r(j)~ i'::>jE1Vl~,NJq.

Corollary 1.2. Let ni be a positive integer, Vj E Jq. Then Vi ~ Ip,

E

n, <

j c Mi NJq

n, jE Mi,f3Jq

Definition. Let K C Jq with [K] = m - p and tr(K) = I o. L e t / ( = J -~ K. Let z be a point-to-set m a p p i n g f r o m / ( into 1o s u c h that

r(j) n ~(j) = ~,

Vj ~ g

and = rn

lT(J)l

-

p

-

2.

Let X = (x, y, s). T h e n X(cr, K, r , p ) will be defined as follows: Vi ~ ~ , Vj E J, x0 = 0, and 3j = j(i): s~(j)i = O.

V i ~ Ip,

] V j E Mi,

I

x~=0.

VjE1VIinK, Vj ~ K,

sij = O.

s~(j)j = 0.

Proposition 1.3. X(tr, K, z, p) is a vertex of L P ( m , n). Proof. One can easily c o m p u t e the remaining c o m p o n e n t s of X(o-, K, r, p): x,j =

l m-p-I

vi4.,

j ~

~,

and Y~

_m-p-2

y/=l

m-p-I or

or yi=0

1

Yi-m_p_l

ViEIp

ViEfp.

By [2, T h e o r e m 8], the non-integer solution X(cr, K, r, p) is a vertex of L P ( m , n).

Example. L e t m = 5, n = 7 , p = 1 . I o = { 1 , 2 , 3 , 4 } , J = {1,2, 3,4, 5 , 6 , 7 } , r ( 5 ) = {1,2}, 9 (6) = {2, 3}, r(7) = {1,4}. M, = {1}, M2 = {2}, M3 = {3, 7}, M4 = {4, 5, 6}. K = {1,2, 3, 4}, [K[ = 4, o'(K) = 1o = {1,2, 3, 4}, j(5) = 1.

153

Monique Guignard/ Plant location problems

T h e k n o w n e n t r i e s o f X(cr, K , ~', p ) a r e ^g

K 0 0

r.{5

0

0

0

0 0 0

0 0

0

0

0

0

0

0

0 0

0 0

0 0 0

0 0

0 0

0

0

x

Y

W e m u s t d e t e r m i n e the o t h e r e n t r i e s

0 Y2 Y3 Y4

Yl 0 Y3 Y4

Y~ Y2 0 Y4

Yl Yz Y3 0

0

0

0

0

0

0

0

0

0

0

Yl Y2 Y3 Y4

Yl 0 0 0

0 y: 0 0

0 0 Y3 0

0 0 0 Y4

0 0

1

0

0

0

0

0

Y4

0 0 y4

Y3 0

0

0

0

X

+ Y4

Yl + Y2

:~Yl=

Y 2 = Y 3 = Y4=31 9

Yl + Y2 + Y3 Y2 + Y3 + Y4

0 1/3 1/3 1/3

1/3 0 1/3 1/3

1/3 1/3 0 1/3

1/3 1/3 1/3 0

0

0

0

0

0

0

0

0

0

0

x Vj~ K,

2/3 2/3 2/3 2/3

1/3 0 0 0

0 1/3 0 0

0 0 1/3 0

0 0 0 1/3

0 0

1

0

0

0

0

;

0

1/3

0 0 1/3

1/3 0

0

0

0

s

x ~ m = I - Y,cJ) = Y,~) = 31. vi~

~-(j),

x~ = 1 - 3 - z _ I,

s~j = z - 31- 32 = o.

0 113 1/3 1/3

1/3 0 1/3 1/3

1/3 1/3 0 1/3

1/3 x/3 1/3 0

1/3 1/3 1/3 0

1/3 1/3 1/3 0

1/3 1/3 0 113

2/3 2/3 213 2/3

1/3 0 0 0

0 1/3 0 0

0 0 1/3 0

0 0 0 1/3

0 0 0 1/3

0 0 0 1/3

0 0 113 0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

x

y

s

Monique Guignard/Plant location problems

154

2. Valid inequalities associated with X(o', K, ~', p) Consider the inequalities x0+yi_-_l,

~, xq

Vi~Ip, VjEf/l~OJq, (*q)

<: 1, Vj ~ Jq

(~,j)

satisfied as equalities by X(~, K, ~', p). Any nonnegative integer combination of these inequalities will be a valid inequality for SPLP(m, n). If the resulting left-hand side coefficients are all congruent to 0 modulo an integer and if the right-hand side is not, one can divide all coefficients by that integer and round down the right-hand side. The result will still be a valid inequality. Multiply every (*ij) inequality by a positive integer ni and every (~j.) inequality by (a - nil where a is an integer larger than hi. The composite inequality will be such that Vi ~ Ip, Yi will have a coefficient ~ie~inJq ni and xii will appear only if j ~ M~, i.e. if i# or(j), with the coefficient nj + (a - ni) = a. Proposition 2.1. If the system

~,

ni=a

ViEIp

has a positive integer solution (ni, a), then E E x'i+E Yi<--m+n-p-q

(F..,q)

-2

is a valid inequality for SPLP(m, n). Proof. Assdme that (ni, a), positive integers, satisfy

~, nj=a

Vi~Ip.

JEMiNJq

Then the composite inequality will read

a[~, ~, xq+ ~, ~ i ] < , v ( m - p ) + ~. ( a - n , ) iEJq i~EJO)

iElp

jEJq

<--a(m - p ) + ( n - q ) a -

~, nj.

J~Jq

Monique Guignard/Plant location problems We know that Vi E Ip, M~ ~ Jq ~ 0 and

L;. n,< Z

JeMinSq

je~fiCISq

155

.,--a

therefore

a<~ns

S6Sct

~

=

ns + ~

]ElfffinSq

i.EMitqSq

nj < 2a

and 1 < ( I / a ) ~i~sq nj < 2, i.e.

[-(l/a) ~j~sq njJ = -2.

Then

~

xii+~pYi<-m+n-p-q

-2

is a valid inequality for SPLP(m, n).

Proposition 2.2. The system

~,

nj=a,

ViEIp

je~nsq

has a positive integer solution. Proof.

a = ~_, n~iesq

~

jE~nsq

nj

is a constant for all iEIp, so that nj, j~Jq, must be such that Vi# i', i, i'EIp:

Zn,=Y JEMiNJa

n,. JEMi,C'IJq

Consider B = LCM{(IMi t3 J~l), i E I,}. Then, set nj = BI(IM~o) f'l Jq[), j ~ J~. ns will be a positive integer. As the sets M~ A Jq form a partition of Jq, all ns are determined in this way, and

a = ~q

lsNnsq~ n s = [ 3 ( m - p - 1 )

is a positive integer.

Corollary 2.3.

F,,,p,,7 is a valid inequality which eliminates X(cr, K, T, p).

Example. Consider again the example of Section 1. Suppose q = 0. We must find a, nj, positive integers such that n 2 "4- n 3 -~- n 4 Jr- n 5 -~- n 6 -~-/I 7 - - (7/ -~ O,

n~

+n3+n4+ns+n6+n7-a=O,

n l q- n 2

nl + n2+ n3

O) (2)

"1-n4 -.1-n5 .-~-n6 -1- n7 - a = O,

(3)

+ n7- a = O,

(4)

Monique Guignardl Plant location problems

156

It is clear that n~ = n2 = n3 + n7 = n4 + n5 + n6. T h u s a -- n~ + na+ n7+ n3+ n7 = nl + 2n3 + 2n7

(1)

= n~ + 2n3 + 2n7

(2)

= 2n~ + n3 + n7

(3)

= 2n~ + n3 + n7

(4)

and finally a = 3n~. Choose/3 = LCM(IM~I) = LCM(1, 2, 3) = 6. Then: j 1 n~ 6 / 1 = 6

2 6/1=6

3 6/2=3

4 5 6/3=2 6[3=2

6

6/3=2

7

6/2=3

and a = 18. The valid inequality F~,p,q reads: X12 + X13 "Jr X14 "~-X15 q- X16 "al-X17 ] + X23 + X24 + X25 + X26 + X27 ] + YJ + Y2+ Y3+ Y4 -----9.

+X21 "l-X31 + X32

"1-XM -~-X35 "~-X36

"~-X41q- X42 -~" X43

-I- X47

For X(tr, K, r , p ) the left-hand side is 7 + 4 •

= 9{3 and F~.p,q is violated.

3. Some facets of SPLP(m, n) Consider now p = 0, q = n - m and Jq = K. F~,p,q becomes: ~ xij<~iYi+m-2. iEK i~r(j)

(Fo, r )

Rather than being derived from the previously treated valid inequalities, these facets could also be generated more directly with n i = 1. H o w e v e r , the simple form of the valid inequalities, and their direct relationship with fractional vertices, may make them important in their own rights, for example in a simplex-based algorithm.

Proposition 3.1.

F~,r is a f a c e t o f S P L P ( m , n).

Proof. We must show that F . , r contains m n + m - n linearly independent integer vertices of S P L P ( m , n), and that there exists at least one integer vector satisfying F~.r as a strict inequality.

Monique Guignard[ Plant location problems

157

(a) (1) L e t X i be defined by x/~ = 1, Vj, Yi = 1, Yk ---- 0 , V k # i. X i satisfies E,,r as an equality:

~_, ~, x,j = m - l = ~fl y~ + m - 2.

jEK i#,r(j)

T h e r e are m such vertices X ~. (2) C o n s i d e r j o < j ~ < j 2 < . . . < jh, such that j~k < jo with t r ( k ) = tr(j ~ and tr(j ~ = cr(j l) . . . . . o'(j~). L e t i ~ ~(jo). T h e n

X i ~ will be defined by xi? = 1,

x~k = O, V k S jo

X{' will be defined by x0~ = x~i, = 1,

Xik = O, V k # jo, jl

X{ h will be defined by xo~ = xij~ . . . . .

x~p = 1,

Xik = O, V k r jo .... jh, and for all X{ t, ! = 0 . . . . . h, all other shipments are m a d e in r o w t~(j~ x~o% = 1, V j S jo . . . . . j~ and Yi = ! ; Y,o% = 1 if tr(j ~ S i, Vi. T h e r e are n(m - 1) such X{. (3) W e w a n t to p r o v e that {xi}, i E I , {X{}, j E J , i S tr(j) are linearly i n d e p e n d e n t by showing that ~,i~i~(+ ~ 9

~

fli~x{=O==>[~i=O

j i~,,~)

0

Vi, Vj: iStrO'), Vi.

L e t us f o r m a matrix w h o s e r o w s are indexed by Yl. . . . . Ym, x , . . . . . Xm,, and w h o s e c o l u m n s are {Xi}, i E I and {X{}, j ~ J , i S tr(j). T h e rows x~.~,-J/r(i)have only one 1 in c o l u m n X i, so that ai ~ 0, Vi. With the same notation as a b o v e , we obtain ~,~ + ~j, + . . . + ~j~ = ~ , + - - . + ~j~ = fl~j~, ..... o'(j h) ~ i so that flis~ = ~ij' . . . . . ~ i p - , = O. The only remaining c o l u m n s are X{ h with i S tr(jh), say X~ ~, X~ i. . . . . X~ !k', iEL It is e a s y to see that

Vjo < jl < . . . < jh with tr(j ~

Vi:

/3ii, =/3/j~ . . . . .

/3/j~k,=/3i,

say f r o m the x~j rows, since they n o w a p p e a r i n d e p e n d e n t l y in equal sums w h o s e other terms are identical. Finally, b y substituting into the y; rows, one must solve the s y s t e m 01...

1

1.

.

1 1... 10

iit=

Monique Guignard!Plant location problems

158

We know that this matrix is nonsingular and that all/3i must be zero. The m + n ( m - 1) = m n + m - n vertices are linearly independent. (b) The vector (x, y), defined by

i = cr(j),jEJ,

I'1 Xii =

otherwise,

y,.=0

Vi,

satisfies F~,K with a strict inequality. (c) F~.K is therefore a facet of SPLP(m, n). Example. m = 3, n = 7, Ml = {1}, M2 = {2, 6, 7}, M3 = {3, 4, 5}. The facet is X 1 2 + X l 3 + X21 + X 2 3 + X31 + X 3 2 ~-~ Y l +

I 2 3

1

1

1 1

1 1 1 1

I I 1 I

I

1

I

I

I

I

I 1

1 1 1 1 1 1

1 I 1 I

1 1 I 1

I 1 I

I I 1 1

31 32 36 37

I

1

1

I

x12 13 14 15 16 17 21 23 24 25

I

Y2 +

1

I 1 1 1

I 1

1 1 I

1 1 1 t

1 1 1 1

1 I 1 I

1. 1

1 1 1

1 1 I 1 1 1

1 1 1 1 1 1 I 1 1 1

l I 1 I

1 1 I 1

1

1 I

1 I 1

1 I

I

1 1 I I 1 1 1 1

1

I1 22 26 27 33 M 35

1

1

1 1

Y3 +

I 1 1

~

1 1

1 1 I

1

xl

x~

x~

x~

x]

x~

1 I

1

x~

x~

1 1 1

1

1 1 1

x~

x~

x]

x~

x1

Jr

~

3

x

Remarks. In the (x, y) space, by varying p, K, and or, one obtains quite a number of vertices of the form X(tr, K, z, p). For instance, for rn = 3, n = 4, one has 36 fractional vertices of the form X ( m K, r, p) vs. 81 integer vertices. Similarly, one obtains (g). m! facets F,,,K. For example, with m = 3 and n = 4, one has 24 facets. Notice that some fractional points are eliminated by several facets. For instance, the fractional solution x=

y=

Monique Guignardl Plant location problems

159

is eliminated by both xt~ + x3~ + x22 + x32 + x~3 + x23 -< y~ + Y2+ Y3+ 1, and X/4-['- XM "q'-X22 "~- X32 "1- X13 -I- X23 ~---Yl -F y2 + y 3 + 1.

On the other hand, each-facet eliminates m " - " fractional vertices with p = 0, and many more with p > 0.

4. Duality gaps and facets We shall now consider a special family of S P L P ( m , n)'s, for which the structure of the cost matrix is such that the continuous and the integer optimal solutions are known explicitly and the duality gaps are very large. For these problems we shall show that a facet of the form F~.K bridges the duality gap. Let tr be a surjection from J into I. L e t c, ti)j be infinite, and let c,.i = c, V i # tr(j). In other words, the shipment costs are either infinite or equal to constant c. L e t jfi = L, Vi E I. We shall denote this problem by SPLP(o-, L).

Proposition 4.1. The relative duality gap f o r SPLP(~r, L ) c a n n o t exceed 89 Proof. A continuous optimal solution will be determined by:

u e J:

Xo-(j)j = 0, Vi E L i r tr(]),

x~i = 1/(m - 1)

and Vi E L

yi = I/(m - 1 ) .

The cost of this solution is zc=m

9

L m-1

c

+ n 9( m - 1 ) . - m-1

mL ---~-n.c. m-1

An integer solution will correspond to two open plants (given that at least one route is prohibited from an open plant by means of an infinite cost), and an optimal solution will, for instance, be defined by: Yil = Y/2

=

1, il, i2 E L and say il < i2

xi, j = 1, V j E 1~i,, xid = 1, V j E Mi, C JQ1,.2,

with a total cost of zl = 2 L + nc.

160

Monique Guignard] Plant location problems

T h e n t h e r e l a t i v e d u a l i t y g a p a will b e = (2L + nc - ((mL/(m

a = (z~ - z r

= (2L - mL/(m

- 1)) + n c ) ) / ( 2 L

+ nc)

- 1))/(2L + n c )

= (2(m - 1)L - mL)/(2(m

- 1)L + (m - 1)nc)

= ( ( m - 2 ) L ) / ( 2 ( m - l ) L + ( m - 1 ) n c ) < ((m - l ) L ) / ( 2 ( m - 1)L) = 89 I n o r d e r to g e n e r a t e a p r o b l e m w i t h a g i v e n r e l a t i v e d u a l i t y g a p a , o n e m u s t choose (m - 2)L = 2a(m

-

I)L + a(m

-

1)nc

or mL

- 2amL

+ 2aL

- 2L = a(m

-

1)nc,

i.e. L = (a(m

-

1)nc)/(m(1

- 2 a ) + 2 ( a - 1))

T h i s m a k e s s e n s e o n l y if m ( 1 - 2 a ) + 2 ( a - 1) > 0, i.e. if m > (2(1 - a ) ) / ( 1 - 2 a ) s i n c e w e k n o w t h a t a < 89 E x a m p l e . F o r a = ~ ( v e r y c l o s e to 89 w e m u s t h a v e m > (2(1 - ~))/(1 - 2 • ~) = (2 • 11)/1 = 22. T a k e m = 23. T h e n L s h o u l d b e e q u a l to - ~ ) + 2(~ ~o -

L -- ( ~ ( 2 2 ) n c ) / ( 2 3 ( 1

1)) = 220 n c .

I f n = 30 a n d c = 1, f o r i n s t a n c e , w e o b t a i n L = 6600. T h e n , z~ = 2 . 6600 + 30 = 13230, and zc = 30 + ~ . 6600 = 6930, so t h a t i n d e e d t h e g a p is = ( 1 3 2 3 0 - 6 9 3 0 ) / 1 3 2 3 0 = ~.

a = (zr - zr

T h e c o s t m a t r i x l o o k s as f o l l o w s : 6600

oo

111

1

1

9I

1 oo

23

ll~176 ~176

1

6600 23

30

1

1 oo

161

Monique Guignard[ Plant location problems

Proposition 4.2. Given a subset K C J such that ( r ( K ) = L IKI = m, F~,K bridges

the gap f o r SPLP(~, L). Proof. A d d F~,K to the constraints o f SPLP((T, L ) and write the corresponding

dual problem: M i n ~ f," Yi + ~ / ~

Max ~ v~ - (m - 2)p = za

cq . xq

I

vj:

=

1

vi unrestricted in sign

i

Vi, Yj:

y~ y, - ~ , jEK

9

Vi:

~,

x/j

-> 0

xq

>- 2 - m

>--0

p

>--0

i~o'(j)

Yi

>- 0

~ w i j + p <<-f,

>- 0

vj -- w~j -- p ~ cq

V j E K, V i ~ o'(j):

x~i

Vj E K:

x~q)j >- 0

VjEJ-~K,

wii

Vi:

x~i

>-0

J

Vj - - Wo'(i)j

vj - wq

<7 Co.(])j

<- cq

The following solution is feasible for the dual: wq=O

Vi, Vj,

P =fi = L , vj=c+L

VjEK,

vj=c

VjUJ-~.K.

Then za=~

(c+L)+

jEK

~

(c)-L(m-2)

jEJ~K

= m ( c + L ) + (n - m ) c - L m + 2L,

that is Zd = 2 L + nc.

Since we know a primal solution with the same objective function value, this is the optimal value for the dual and there is no more duality gap.

162

Monique Guignard/ Plant location problems

References P e r t i n e n t r e f e r e n c e s are either the f o l l o w i n g p a p e r s , or c a n be f o u n d r e f e r e n c e d in them. [1] E. Balas and M. Padberg, Set partitioning, in: B. Roy, ed,, Combinatorial programming: methods and applications (D. Reidel Publ. Co., Dordrecht, 1975). [2] G. Cornuejols, M. Fisher, and G.L. Nemhauser, On the uncapacitated location problem, Annals of Discrete Mathematics 1 (1977) 163-177. [3] M. Guignard and K. Spielberg, Algorithms for exploiting the structure of the simple plant location problem, Annals of Discrete Mathematics 1 (1977) 247-271. [4] M. Guignard and K. Spielberg, A dual method for the mixed plant location problem, with some side constraints, Mathematical Programming 17 (1979) 198-228. [5] J. Krarup and P. Pruzan, Selected families of discrete location problems, Part III: The plant location family, University of Calgary Working Paper No. WP. 12.77. [6] M. Guignard and K. Spielberg, A direct dual approach to a transshipment formulationfor multi-layer network problems with fixed charges, Wharton School Department of Statistics, Report No. 43, August 1979.

Mathematical Programming Study 12 (1980) 163-175. North-Holland Publishing Company

BALANCED CLAUDE

MATRICES

AND PROPERTY

(G)

BERGE

Universit~ Pierre et Marie Curie, Paris, France Received Received 25 January 1979 Revised manuscript received 5 September 1979 Let A be a (0, 1)-matrix with n rows and m columns, considered as the incidence matrix of a hypergraph H with edges Et, E2. . . . . E, (the columns) and with vertices xl, x2. . . . . x, (the rows). H is called a balanced hypergraph if A does not contain a square sub-matrix of odd order with exactly two ones in each row and in each column. In this paper, we prove more "minimax" equalities for balanced hypergraphs, than those already proved in Berge [1], Berge and Las Vergnas [3], Fulkerson et al. [7], Lovfisz [12]; in fact, the known results will follow easily from our main theorem.

Key words: Balanced Hypergraphs, Normal Hypergraphs, Zero-One Matrices, Minimax Theorems, Integral Polyhedra.

1. General Definitions L e t A b e a (0, 1)-matri• w i t h n r o w s a n d m c o l u m n s c o n s i d e r e d h e r e as t h e i n c i d e n c e m a t r i x o f a h y p e r g r a p h H . F o r p E N", q E N m, w e c o n s i d e r t h e following coefficients:

v ( H ; p, q ) = max{(q, z) I z E Z m, z - 0, A z <- p}, ~-(H; p, q ) = min{(p, t) I t E Z", t >- 0, tA >- q}, ~-*(H; p, q ) = min{(p, t) I t ~ R", t >- O, tA >- q}. C l e a r l y , v ( H ; p, q) <- ~'*(H; p, q ) --< r ( H ; p, q). T h e c o e f f i c i e n t v ( H ; p, q) is t h e m a x q - v a l u e o f a p - m a t c h i n g , w h i l e r ( H ; p , q ) is t h e m i n p - v a l u e o f a q t r a n s v e r s a l in H . Some properties of H, which have been considered under different names by different authors, can be rephrased as follows: H is normal (Lov~isz; " p e r f e c t m a t r i c e s " in P a d b e r g [14]) iff

v ( H ; 1 , q ) = r ( H ; 1 , q)

for allqEN

m.

H has t h e Menger property ( S e y m o u r , S c h r i j v e r ) iff

v ( H ; p, l) = ~-(H; p, 1)

f o r all p E N".

h is paranormal ( F u l k e r s o n , L e h m a n , S a k a r o v i t c h , etc.) iff ~ ' * ( H ; p , l) = ~ ' ( H ; p , 1)

f o r all p E N " . 163

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c. Berge[Balancedmatrices and property (G)

Clearly, the Menger property implies paranormality, but the converse is not true. Equivalent formulations exist in the literature. For instance, H is "paranormal" means that every vertex of the "transversal" polytope {t I t ~ R " , t ~ O ,

tA>-l}

has only integral coordinates. Also H is "normal" means that every vertex of the "matching" polytope

{z [ z ~ R m,z >-O, Az <= 1} has only integral coordinates. We shall say that H has the K6nig property iff v(H; 1, 1) = r ( H ; 1, 1). Clearly, if H is normal, or if H has the Menger property, then H has also the K6nig property. Numerous examples of hypergraphs having one of the above mentioned properties exist in the context of Graph Theory (cf. [16]). They appear most often as a "minimax" theorem, expressing the duality principle for some linear program in integers. The other classical definitions needed in this paper are the following. If H = ( E I , E2. . . . . Era) is a hypergraph, its vertex-set is X = U Ei; its rank is r(H) = max[Eil; its anti-rank is s ( H ) = minlE~[; they are both positive integers. H ' is a partial hypergraph of H if it is obtained by removing edges (and all the vertices which become isolated). HA is the subhypergraph induced by A C_X if its edges are the non-empty sets to the form E~ f3 A. Also, H/A is the section hypergraph of H by A, i.e. its edges are all the Ei which are contained in A. For u = (ul, u2..... un) E N n, we denote by H u a hypergraph obtained from H by replacing each vertex xi of H by a set Xj with cardinality ui and each edge E~ by UxjE~, Xj. A set T is a transversal set of H if TAE~#Y

(i = 1,2 ..... m).

The set of all minimal transversal sets of H is denoted by Tr H ("transversal hypergraph"). So, z ( H ; p , 1) can be described as the minimum weight of a transversal set. The number r ( H ; 1, 1), more often denoted by z ( H ) = min{ITI; T ~ Tr H}, is the transversal number of H, and v(H; 1, 1) more often denoted by v(H) is the matching number of H. A well-known result of Lehman-Fulkerson [6, 10] is: Lemma. (Lehman-Fulkerson lemma). Let p E N", w ~ N", and let H be a hypergraph with vertices Xl, x2..... x,. Then H is paranormal if and only if

C. Berge[Balanced matrices and property (G)

165

n

~'(H ; p, l)~'(Tr H ; w, 1) <--~, p, wi i=I

for all p E N", w ~ N L From this result, it follows that H is paranormal if and only if T r H is paranormal. H is semi-normal (Lov~sz) if for every subset A C X, the section hypergraph H/A has the K6nig property. Clearly, if H has the Menger property, or if H is normal, then H is semi-normal.

2. The property (G) Let H be a hypergraph of n vertices with anti rank s(H); denote by or(H) the maximum number of pairwise disjoint transversal sets of H. Clearly,

or(H) <--s(H). We shall say that H has the property (G) if

o'(HA) = S(HA)

(A C X).

We shall say that H has the strong property (G) if

o-(H") = s ( H u) (u E N"). For instance, if H is the dual of bipartite multigraph G with edges el, e2..... en, then H u is the dual of a bipartite multigraph G' obtained from G by multiplicating each edge ej by uj. By a theorem of Gupta [8], we have Q(G') = ~(G'), hence tr(H ~) = s(H"). So, the dual of a bipartite multigraph has the strong property

(G). Proposition 1. A hypergraph H has property (G) if and only if Tr H is seminormal. Proof. Tr H is semi-normal if and only if, for every A C X, the hypergraph ( T r H ) / A satisfies the K6nig property. Since ( T r H ) / A = T r ( H A ) (unless (Tr H ) / A does not exist) this is equivalent to v(Tr HA) = ~-(Tr HA). Note that ~-(Tr HA) is equal to the minimum size of an edge of HA, SO ~-(Tr HA) = s(HA). On the other hand, v(Tr HA) is the largest integer k such that there exist k pairwise disjoint transversal sets of HA, so v(Tr HA) = or(HA) and the proposition follows.

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C. Berge/Balancedmatrices and property (G)

Proposition 2. A hypergraph H has the strong property (G) if and only if Tr H has the Menger property. Proof. Let H be such that Tr H has the Menger property,

v ( T r H ; p , 1)=~-(TrH;p, 1) ( p E N ~ ) .

(1)

For any integer k, each of the following inequalities is equivalent to the consecutive one: ~-(Tr H ; p, 1) - k,

~, pj>-k (i<_m), xi~Ei s(H p) >--k. Hence

s(H p) = ~,(Tr H ; p, 1). On the other hand, H p admits k disjoint minimal transversal sets iff there exist k transversal sets of H, say T~, T2..... Tk E Tr H, such that:

I{i [xjET~}l~--pj

(j<=n).

Since this is equivalent to v(Tr H ; p , 1)=> k, we have

tr(HP) = v(Tr H ; p, 1). Thus, (1) is equivalent to:

o,(H p) = s(SP),

(2)

which means that H has the strong property (G). Corollary. Every hypergraph with the strong property ( G) is paranormal. Proof. From Proposition 2, a hypergraph H with the strong property (G) has a transversal hypergraph H ' = T r H which has the Menger property. By the duality principle of linear programming, we have

v(H'; p, 1) =< ~'*(H'; p, 1) =<~'(H'; p, 1). Since H' has the Menger property we have also ~'*(H'; p, 1) = 7(H'; p, 1). From the Lehman-Fulherson lemma, it follows that H is paranormal. Remark. The converse of the above corollary is not true. For instance, the hypergraph H = {abc, cde, era, bdf, ad, be, cf} is not paranormal; however, s(H) = 2, o,(H) = 1, so that H has not the property (G).

C. Berge[Balanced matrices and property (G)

167

We shall see in the next section that if all partial hypergraphs of H are paranormal, then H has the strong p r o p e r t y (G).

3. Minimax theorems for balanced hypergraphs A h y p e r g r a p h H is balanced if e v e r y odd cycle has an edge containing three vertices of the cycle, i.e. if the incidence matrix of H does not contain an odd square submatrix of the following form: /1100 -.. 0 ~ 0110 ... 0 0011 ... 0 00... 10...

11 01/

Clearly, if H is balanced, then a partial hypergraph, or a sub-hypergraph, or the dual h y p e r g r a p h of H, is also balanced; furthermore, we h a v e

v(H) = r ( H ) ,

q(H) = A(H)

(see [1]). To set m o r e properties for balanced h y p e r g r a p h s we shall show the following result:

Theorem 1. For a hypergraph H = (El, E2 . . . . . Era) on X = {xl, x2. . . . . x~}, the following conditions are equivalent: (1) H is balanced; (2) every partial hypergraph H' C H has the strong property (G); (3) every partial hypergraph H' C_H has a transversal hypergraph Tr H' with the Menger property; (4) every partial hypergraph H' C_H has a transversal hypergraph Tr H' which is paranormal ; (5) every partial hypergraph H' C H is paranormal; (6) every partial hypergraph H' C H has the Menger property; (7) every partial hypergraph H' has a transversal hypergraph T r H' which is semi-normal; (8) every partial hypergraph H' C_H has the property (G). Proof. ( 1 ) ~ ( 2 ) . L e t H be a balanced hypergraph of order n, and let p ~ N " . Since H p is also balanced, we h a v e only to show that e v e r y balanced hypergraph H = (El, E2 . . . . . Era) satisfies

tr(H) = s ( n ) = min[Eil

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C. Berge/ Balanced matrices and property ( G)

Let k = min[Ei[, and let (S,, 82 ..... Sk) be a partition of the vertex-set X into k classes. Denote by k(i) the number of classes which meet Ei. If k(i) = k for all i, all the classes are transversal sets of H, and the proof is achieved; otherwise, there exists an index j _-<m with k(j) < k. Since kfj) < k _-< [/~[, there exists a class Sp such that [Sp n Ej{--> 2. Furthermore, there exists a class Sq such that

Is~n~l=0. The subhypergraph Hspnsq is balanced, and therefore has a 2-coloring (S~, S~). Put S~=S~ if irSp, q. The partition (S~,S~ ..... S'k) determines as above new coefficients k'(i), and

k'(i) > k(i)

(i <=m),

k'(j) = k(j) + 1. With this method, it is always possible to improve a partition (S~, S~. . . . . S~) until we have k ' ( i ) = k for all i; then we have a partition of X into k = s ( H ) transversal sets, and the proof is achieved. (2) ~ (3). This follows from the Proposition 2. (3) ~ (4). This follows from the general relation:

v(H'; p, 1) _-
p, 1) _-
(4) ~ (5). This follows from the L e h m a n - F u l k e r s o n lemma. (5) ~ (6). Let H be a hypergraph whose partial hypergraphs are paranormal, and assume there exists a partial hypergraph of H which has not the Menger property (per absurdum). Let H ' be the partial hypergraph of H which does not satisfies the Menger property and with the least number of edges. Let n be order of H ' and let m be the number of edges of H ' . Let p be the integer vector with (p, 1> minimum such that

(i)

v(H'; p, 1) < ~-*(H'; p, 1) = ~(H'; p, 1).

Let z be an optimal p-matching of H, that is a vector of the p-matching polytope:

{z [ z E R m , z >-O, A z < p } such that ~ zi be maximum. Since z has not all its coordinates integral, by (i), let z,, say, the smallest non-integral coordinate of z. By the minimality of (p, 1>, we have (ii)

0 < z, < 1.

From the minimality of H ' , we know that the partial h y p e r g r a p h / ~ = H ' - E , ,

C. Berge/Balanced matrices and property (G)

169

with incidence matrix fi~, has the Menger property. Thus,

is attained b y an integral vector a = (u2, uj . . . . . urn). On the othor hand, ~ = (z2, z3. . . . . z , ) is also a fractional p-matching o f / ~ , so that m

~'*(H'; p, 1 ) - z l = ~ i =2

m

zi -------~ui = ~'*(H; p, 1). i=2

But z * ( H ' ; p, 1) and z*(/~; p, 1) are both integers, and it follows from (ii) that: (iii)

T*(H'; p, 1) _--<~-*(/~; p, 1).

Since (0, u2, u3. . . . . urn) belongs also to the p-matching polytope of H ' , we obtain: (iv)

T*(H'; p, 1)_--<~ ui = ~-*(lq; p, 1). i=2

Comparing (iii) and (iv), we see that (0, u2, u3. . . . . urn) is an optimal p-matching of H ' ; since all its coordinates are integral,

v(H'; p, 1) = z * ( H ' ; p, 1")= "r(H'; p, 1). This is a contradiction with (i), and achieves the proof. (6) ~ (1). If H satisfies (6) and is not balanced, there exists an odd cycle, say ( X l , E l , X2, E 2 . . . . . X2k+l, EER+I,XO, with no El containing three xj's of the sequence. Take H ' -- (El, E2 . . . . . E2k+l), and

PJ=

1 for j = 1,2 . . . . . 2 k + l , 0 otherwise.

We have

v(H'; p, 1) = k, ~-(H'; p, 1) = k + 1. This contradicts (6) and achieves the proof. (3) ~ (7). The Menger property implies the semi-normality. (7) ~ (8). This follows from Proposition 1. (8) ~ (1). If H satisfies (8) and is not balanced, then there exists an odd cycle, say (xl, El, ..., EEk+b X0, with no Ei containing three x f s of the sequence. Take H ' = ( E I , E2 .... EEk+O, A ={Xl, X2. . . . . X2R+~}. We have s ( H ~ ) = 2 , tr(H~) = 1. This contradicts (8) and achieves the proof. Corollary 1. Every balanced hypergraph has the Menger property; that is, the

170

C. Berge/ Balanced matrices and property (G)

minimum p-value of a transversal set is equal to the m a x i m u m cardinality of a p-matching.

This is a result of Fulkerson et al. [7]. Corollary 2. The transversal hypergraph of a balanced hypergraph has the Menger property. This is a partial answer to a problem raised by Fulkerson et al. [7], who have noted that the transversal hypergraph of a balanced hypergraph is not necessarily balanced, but asked if it may have similar properties.

4. Minimax theorems for balanced hypergraphs which follow from the normality

Let H be a hypergraph; its chromatic index q ( H ) is the least number of colors needed to color the edges so that two intersecting edges have different colors. Let A(H) be the maximum degree, that is the maximum number of edges having one point in common; H is called a normal hypergraph if every partial hypergraph H ' of H satisfies q ( H ' ) = A(H'). Several characterisation of normal hypergraphs have been given by Lovdsz [11], Padberg [13, 14] and by Chwitai [4, 5]. The main results have been summarized by Lovfisz in a very simple theorem, that we shall use (in a slightly different context) to get more minimax theorems. Lenuna (Lovdsz [11]). Let H = ( E I , E2 . . . . . Era) be a hypergraph on X = {xl, x2 . . . . . x,}, and let y = (Yl, y2. . . . . y,,) E N ~. Then the hypergraph ffI obtained from H by multiplicating each Ei by yi is also normal. It suffices to show that /~ = ( E I , E~, Ez, E3 . . . . . Era), with E~ = E~, satisfies q(/-)) --- A(/~). Put q ( H ) = A ( H ) = q. Case 1. In H, the edge E~ contains a vertex x with degree du(x) -- A ( H ) . Then, A(/~) = q + 1, and we have A(fft) <=q ( f I ) <--_q ( H ) + 1 = q + 1 = A ( f I ) .

Hence, q(/4) = A(/~). Case 2. The edge E1 does not contain a vertex with degree A(H), i.e. dn(x) <=q - 1

(x E EO.

Let denote by (I) the color received by the edge E~ in an optimal q-coloring of the edges of H ; each vertex of H with maximum degree belongs to an edge of color (1).

C. Berge ! Balanced matrices and property ( G )

171

Denote by Ht the family of edges of H having color (1) and different from E~. So, A ( H --Hi) = q -- 1. Since H is normal, q ( H - H ~ ) = q - 1; therefore, we can color with q - 1 colors the edges in H - H~, and with one new color for HI + El, we obtain a q-coloring o f / ~ . Hence q ( ~ ) __0, Ay -< 1} has (0, D-coordinates ; (3) every vertex of the matching polytope has integral coordinates. (4) v(H; 1, q) = I"*(H; 1, q) for all q E Nm; (5) v(H; 1, q) = ~'(H; 1, q) for all q E N m ; (6) every partial hypergraph H ' has the K6nig property. Proof. ( 1 ) ~ ( 2 ) . Let z be a vertex of the polytope Q; since z is determined by a system of equalities with integral coefficients, its coordinates are rational, and there exist integers k, pl, p2 ..... Pm >-0 so that kz = (Pl, P2. . . . . P~). In the hypergraph H obtained from H by multiplicating each edge E~ by p~, we have d~(x~) = ~, Pi = (a j, kz) = k(a s, z) <-_k. E~xj

Hence A(/4) _---k, and, by the lemma, q ( H ) _-
fl

if one copy of E, is in/-I~,

l0 otherwise.

The vector y~ = (y~, y~.... , Ym), with (0, 1)-coordinates, belongs to Q; furthermore l~yk -k

1 = ~ ( P l , P 2 . . . . . Pm)=z.

Since z is a vertex of Q,

C. Berge/ Balanced matrices andproperty (G)

172 y l = y2 . . . . .

yk = Z.

This shows that z is a vector with (0, D-coordinates. (2) ~ (3). Obvious. (3) ~ (4). Since max{(q, y) [ y E Q} is attained by a vertex of the polytope Q, we have

v ( H ; 1, q) = ~-*(H; 1, q). (4) ~ (5). Put Ql = {z [ z E Q, (z, q) -- maxy~ o (y, q)}. Q1 being a face of the polytope Q, there exists a r o w - v e c t o r t~1 of the incidence matrix A such that

z E Ql ~ (a i',z) = 1. It follows f r o m (4) that each matching of H with m a x i m u m q-value covers the vertex xjl. Put

ql=

qi -- 1 ff xi~ E E i , qi otherwise.

Thus

v(H; 1, ql) = v ( H ; 1, q ) - 1. As above, there exists a vertex xj2 of H and a vector q2 = (ql,2 q22. . . . . q2) such that v ( H ; 1, q2) = v ( H ; 1, q l ) _ 1. We can continue, to define a sequence tr = (xiz, xi2. . . . . xjk), until we h a v e

v ( H ; 1, qk) = O. The vector t = (h, t2. . . . . t n ) , where ti is the n u m b e r of a p p e a r a n c e s of xj in the sequence or, is a q-transversal of H such that:

(t, 1 ) = s

j=l

ti = k = v ( H ; 1, q).

H e n c e t is a q-transversal of minimum value and z ( H ; 1, q) = v ( H ; 1, q). (5) ~ (6). L e t H ' be a partial h y p e r g r a p h of H, and put

{1 qi =

if Ei E H', otherwise.

The vector q = (q~, q2 . . . . . qm) satisfies

C. Berge/ Balanced matrices and property (G)

173

v ( H ; 1, q) = v(H'), r ( H ; 1, q) = ~-(H').

Thus, (5) implies v(H') = "r(H'). (6) ~ ( 1 ) . It su1~ces to show that a hypergraph H which satisfies (6) is such that q ( H ) = A ( H ) . L e t / ~ be a hypergraph whose vertices are the matchings of H, and where an edge/~i denotes the set of all matchings of H containing El. Clearly,/~i N/~i = )ff if and only if El A E i # ~ . Since H satisfies (6), it has the Helly property, hence v(~) = a(H).

Also, q(fI) = r(H), r(ff-I) = q ( H ) , A (if,I ) = v(H).

Hence, /~ is normal; since we have already shown that ( 1 ) ~ (6), we have also v(H) = r ( H ) , that is q(H)=A(H).

This achieves the proof. Theorem 3. L e t H = (E,, E2 ..... Era) be a hypergraph on X with incidence matrix A. For J C_{1, 2 .... , n}, we denote by A J the matrix obtained f r o m A by deleting the rows whose index is not in J. For I C_{1, 2 ..... m}, we denote by AI the matrix obtained from A by deleting the columns whose index is not in I. The following conditions are equivalent: (1) H is balanced; (2) All the vertices of the matching polytope {y ] y ~ Rm, y _>0, AJy <=1} have integral coordinates;

(3) V(HA; 1, q) = **(HA; 1, q) for all q ~ N m and all A C_X ; (4) V(HA ; 1, q) = T(HA ; 1, q) for all q E N m and all A C_X. This follows immediately from Theorem 2, and from the fact that H is balanced if and only if HA is normal for all A C_X. Corollary 1. Let H be a balanced hypergraph; then v ( H ; l, q) = T(H; 1, q), i.e. the m a x i m u m q-value of a matching of H is equal to the minimum value of a q-transversal of H.

174

C. Berge/Balanced matrices and property (G)

(This is a result o f F u l k e r s o n et al. [7].) Corollary 2. For all J the vertices of the matching polytope: {y [ y E R m , y =>0, AJy <- 1}

have integral coordinates if and only i[ [or all the vertices o f the transversal polytope {z I z E R ~,z >--O,zA, >=1} have integral coordinates. This follows f r o m T h e o r e m 3 and T h e o r e m 1. R e m a r k . A b a l a n c e d h y p e r g r a p h H satisfies

v ( H ; p, q) = r ( H ; p, q)

(1)

f o r p = 1 and q E N ~' ; or f o r p E N ~ a n d q = 1. H o w e v e r , (1) d o e s not hold f o r e v e r y pair (p, q); f o r H = (ad, bd, cd, abcd), p = (2, 2, 2, 3) a n d q = (1, 1, 1, 2), we see that t -_ tl t2, I2, I2, 89 is a fractional q - t r a n s v e r s a l , and z = (89 89 89 a2) is a fractional p - m a t c h i n g with (p, t) = (q, z), so that z * ( H ; p, q) = 9. Also,

v ( H ; p, q) = 4, r ( H ; p, q) = 5. Hence

v ( H ; p, q) < ~*(H; p, q) < r ( H ; p, q). T h u s (1) is not true f o r H. In fact, no b a l a n c e d h y p e r g r a p h s , e x c e p t the u n i m o d u l a r o n e s , satisfy (1) f o r all p E N n, and q E N m. T h e a b o v e c o n s i d e r a t i o n s suggest the following p r o b lem: D e t e r m i n e all pairs (p, q) with p E N n a n d q E N m such that v ( H ; p, q) = ~'(H; p, q) holds f o r all b a l a n c e d h y p e r g r a p h s H.

References

[1] C. Berge, "Balanced matrices", Mathematical Programming 2 (1972) 19-31. [2] C. Berge, "Sur une extension de la th6orie des matrices bi-stochastiques", in: Dall'Aglio, ed., Studi de probabilita, satistitica e ricerca operativa, in onore di G. Pompilj. (Oderisi-Gubbio, Rome 1970) 475-482. [3] C. Berge and M. Las Vergnas, "Sur un th6or~me du type K6nig pour hypergraphes", Annals of the New York Academy of Sciences 175 (1970) 32-40. [4] V. Chvfital, "On certain polytopes associated with graphs", Journal of Combinatorial Theory 18(B) (1975) 138-154.

C. Berge/ Balanced matrices and property (G)

175

[5] V. Chv~tal, "On the strong perfect graph conjecture", Journal of Combinatorial Theory 20(B) (1976) 139-141. [6] D.R. Fulkerson, "Blocking and anti blocking pairs of polyhedra", Mathematical Programming 1 (1971) 168-194. [7] D.R. Fulkerson, A.J. Hoffman and R. Oppenheim, "On balanced matrices", Mathematical Programming Study 1 (1974) 120-132. [8] R.P. Gupta, "An edge-coloration theorem for bipartite graphs of paths in trees", Discrete Mathematics 23 (1978) 229-233. [9] A.J. Hoffman, "A generalisation of max-flow min-cut", Mathematical Programming 6 (1974) 352-359. [10] A. Lehman, "On the width-length inequality", mimeo (1965). [11] L. Lovasz, "Normal hypergraphs and perfect graph conjecture", Discrete Mathematics 2 (1972) 253-267. [12] L. Lovhsz, "On two mini-max theorems in graphs", Journal of Combinatorial Theory 21(B) (1976) 96-103. [13] M.W. Padberg, "On the facial structure of set packing polyhedra", Mathematical Programming 5 (1973) 199-215. [14] M.W. Padberg, "Perfect zero-one matrices", Mathematical Programming 6 (1974) 180-196. [15] M.W. Padberg, "Almost integral polyhedra related to certain combinatorial optimization problems", Linear Algebra and Appl. 15 (1976) 69-88. [15] M.W. Padberg, "Almost integral polyhedra related to certain combinatorial optimization problems", Linear Algebra and its Applications, 15 (1976) 69-88. [16] A. Schrijver, Fractional packing and covering, in: Packing and covering, (Mathematisch Centrum, Amsterdam, 1978) 175-248.

Mathematical Programming Study 12 (1980) 176-196. North-Holland Publishing Company

DUAL INTEGRALITY IN b-MATCHING PROBLEMS W. P U L L E Y B L A N K The University of Calgary, Calgary, Alberta, Canada

Received 24 June 1977 Revised manuscript received 20 July 1978 We investigate the problem of when a b-matching problem with integer edge costs has an integer optimal dual solution. We introduce the concept of b-bicritical graphs, give a characterization of them and show that these play a pivotal role in determining when there exists an integer optimal dual solution. Key words: b-Matching, Matching, Dual Integrality, Bicritical, b-Critical, Duality.

I. Introduction The weighted b-matching problem is a well solved class of integer programming problems. For such a problem we know a polynomially bounded algorithm which solves the problem, we know a good optimality criterion and we know the unique minimal set of inequalities necessary and sufficient to define the convex hull of the feasible solutions. In fact these are all related via the duality theory of linear programming. In this paper we study the dual linear programming problem and obtain integrality results. There are several reasons why we are interested in such theorems. First, when a b-matching problem has an integral optimal dual solution, the duality theorem of linear programming gives a combinatorial min-max theorem. An example of this is the famous K6nig theorem which states that the maximum cardinality of a matching in a bipartite graph is equal to the minimum cardinality of a set of nodes that meets e v e r y edge. One way of proving this theorem is by showing that the associated linear program always has a 0-1 valued optimal dual solution. A second reason for studying this problem is that the problem of finding an optimal integer solution to the dual of a b-matching problem is well-known to be very difficult. For example in the case that all edge costs are 1 and all degree constraints are 2, the set of nodes given a dual variable of 1 in 0-1 valued solution to the dual problem is a set of nodes that covers all the edges of the graph. Thus the problem of finding a minimum integer solution is polynomially equivalent to the problem of determining whether or not a graph has a set of k nodes that cover the edges. This problem is well-known to be NP-complete (see Aho et al. [1]). We elaborate on consequences of this viewpoint of the results presented here in another paper [8]. In Section 2 we present the basic terminology used throughout the paper. In 176

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Section 3 we describe the weighted b-matching problem and give some fundamental results that are used in the rest of the paper. When determining whether a b-matching problem has an integer optimal dual solution, the set of primal constraints that we use is very important. We observe in Section 3 that if enough redundant primal constraints are added to the problem, then there always exists an integer optimal dual solution. Thus through the bulk of this paper we require that the set of primal constraints used be essentially minimal, that is, it should correspond to the set of facets of the matching polytope. In actual fact, for ease of development, we do use a set of primal constraints which may include some non-facets, but in Section 7 we show how this may be restricted to the case in which only dual variables corresponding to facets are permitted. In Section 4 we introduce the concept of b-bicritical graphs. If we are given a graph G(V, E, ~) and a vector b -- (bi; i E V) of positive integral degree constraints, we say that x = (xj: j E E) is a perfect b-matching if x is an assignment of nonnegative integers to the edges of G, such that for each node i, the sum of the xj on the edges j incident with i is exactly equal to b~. Then we say that G is b-bicritical if G is connected, I VI > 2, b; -> 2 for all i E V, and whenever we reduce any bi by 2, there then exists a perfect b-matching. These graphs are shown to play a central role in determining when a b-matching problem has an integer optimal dual solution. This is the subject of Sections 5 and 6. In Section 5 we consider the case in which all edge costs are equal to 1 and in Section 6, we consider the case in which the edge costs are arbitrary integers. It is known that for any b-matching problem in which the edge costs are integers, there exists an optimal dual solution for which all dual variables are integer or half integer. We call such a dual solution discrete. Theorem 5.5 shows that if G is b-critical and edge costs are 1, then the only optimal discrete dual solution is obtained by letting every node have a dual variable of 89and letting all other dual variables be 0. Theorems 5.9 and 6.2 show that when no integer optimal dual solution exists, the set of dual variables that receive fractional values in an optimal dual solution can be restricted to the node set of a b-bicritical subgraph of the graph (which also satisfies some additional properties).

2. Terminology A graph G is an ordered triple (V, E, ~b) where V is a finite set of nodes, E is a finite set of edges and ~b is an incidence function. For any j ~ E, g,(j) is the set of two nodes incident with the edge j. For any S _C V we let 8(S) denote the set of edges incident with exactly one node of S. That is, 8(S) -- {j ~ E: Irk(j) N SI = 1}. If S consists of a single node v, then we write 8(v) for 8({v}) and so B(v) is the set of edges incident with v. For any S _C V, we let y(S) denote the set of edges having both ends incident with S. That is,

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~,(s) = (j ~ E: t~(i) r S}. For any S C_ V, we let G[S] denote the subgraph of G induced by S, That is,

6 [ s ] = (s, y(s), d ) where ~' is the restriction of 0 to y(S). For any vector x = (x~: j ~ J) and for any K ~ J we let x(J) denote ~,ier x~. Thus for example if x is a vector indexed by the edges of G, and S is a subset of nodes, then x(8(S)) is the sum of the values over those edges having one end in S, x(y(S)) is the sum of the values over those edges having both ends in S. For any S C_ V, N(S) is the set of nodes of V - S which are adjacent to a node of S. Thus ~ E N(S) if and only if there is some j E E such that ~(j) - S = {~}. We abbreviate N({v}) by N(v). Thus N(v) is simply the set of nodes of G adjacent to v. Finally let be any set of subsets of V. Then for any edge L ~(J) denotes those members of ~ that contain both nodes incident with j. Thus 9(j) = {s ~ ~: ~,(j) c_ s}.

3. The weighted b-matching problem Let G =(V, E, t~) be a graph, let b = (b~: i E V) be a vector of positive integral degree constraints and let c = (cF j E E) be a vector of arbitrary real edge costs. The b-matching problem is to find a vector x = (xF j ~ E) such that xi is a nonnegative integer

for every j E E,

x(8(i)) < b~ for every i E V,

(3.1) (3.2)

cx is maximized subject to these conditions.

A vector satisfying (3.1) and (3.2) is called a b-matching. If a b-matching x satisfies x(6(i)) = bi for all i E S C_ V, then we say that x is a perfect b-matching on S. The matching polytope P(G, b) is defined to be the convex hull of the set of b-matchings. (Note that since the zero vector is always a b-matching, P(G, b) is never empty.) We are able to give a minimal set of linear inequalities necessary and sufficient to define P(G, b) (Edmonds and Pulleyblank, see Pulleyblank [7]). In order to describe these, we first introduce some terminology. Let S _C V be such that b(S) is odd. A near perfect matching (np-matching) of G[S] deficient at node i ~ S is a b-matching of G[S] which satisfies

x(8(i)) = b~- 1, x(~(v)) = b~ for every v E S - {i}.

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That is, an np-matching is a b-matching of G[S] which is "as close as possible" to being a perfect matching on S. We say that G is b-critical if for each node i there exists an np-matching of G deficient at i. Note that if a graph G is b-critical, then G is connected and b ( V ) is odd. (A simple example of a b-critical graph is a triangle with bi = 3 for every node i.) We let = {S C V: ISI -> 3 and

G[S] is b-critical}.

(3.3)

Theorem 3.1 (Edmonds and Pulleyblank, see Pulleyblank [7]). P(G, b) = {x ~ RB: X_>0,

(3.4)

X(8(i)) <--bi for all i E V,

(3.5)

x ( 7 ( S ) ) <- 89

(3.6)

- 1) f o r every S E ~}.

This set of inequalities may not in fact be minimal, but it is more convenient to work with here, and our results are easily shown to hold when we have a minimal set of inequalities. This is the subject of Section 7. By virtue of Theorem 3.1, we can now Consider the b-matching problem as the linear program: maximize cx subject to (3.4), (3.5), (3.6). The dual linear program is minimize

~ biyi + ~, qsYs, iE V

SE~

Yi -> 0

for all i E V,

Ys->0

for a l l S E ~ ,

y(~b(])) + y(~(j)) -> cj

(3.7) for all j E E,

where for any S E ~, qs = I2(b(S) - 1). This linear programming problem (3.7), henceforth called the dual problem, is the subject of our study. If the vector c of edge costs is integer valued, then in some cases we can find an optimal solution y* to the dual problem which is integer valued. In fact we can always find an optimal dual solution which satisfies the following discreteness property.

Theorem 3.2 (Edmonds, see Pulleyblank [7]). If c is integer valued, then there exists an optimal dual solution y* satisfying y~ is integer valued

f o r all S E ~

yi* is integer or half integer valued

(3.8)

for all i E V.

(3.9)

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We call a dual solution that satisfies (3.8) and (3.9) a discrete dual solution. Note that every integer dual solution is a discrete dual solution. In fact Theorem 3.2 is an immediate consequence of Edmonds' b-matching algorithm. It solves a b-matching problem by finding a b-matching and a dual solution which satisfy the complementary slackness conditions for optimality. Provided that we initialize the algorithm in the proper fashion, if we have integer edge costs, then the optimal dual solution found by the algorithm will be a discrete optimal dual solution. The complementary slackness conditions of linear programming which are satisfied by optimal matchings and dual solutions are the following. Theorem 3.3. A b-matching x and a dual solution y are optimal if and only if x(8(i)) = bi x('y(S)) = qs

for every i E V such that Yi > O, for every S E ~ such that Ys > O,

y(O(j)) + y(~(j)) = ci for every j E E such that x i > O. When studying dual integrality, our set of primal constraints is very important, since we have a dual variable available for each such primal constraint. Thus although the presence of superfluous primal constraints does not affect the set of solutions to the b-matching problem, the presence of corresponding superfluous dual variables may enable us to find an integer optimal dual solution when none exists if we are limited to dual variables corresponding to essential primal constraints. Let ~ = {S _C V}. Then for any S ~ ~, for any b-matching x, we have x( ?/(S)) <--[89

where for any real number t "[t]" denotes the greatest integer no greater that t. This is true since for any b-matching x, b(S) >- ~, x(8(i)) = 2x(~/(S)) + x(8(S)) i~s

and so x(y(S))<_ 89 But x ( y ( S ) ) is integer valued and the result follows. Note that if S E 2, then x ( y ( S ) ) < - 8 9 1) is the same constraint as x(y(S))<[89 Thus it follows from (3.4) that P(G, b) = {x E RE: x satisfies (3.4), (3.5) and x ( y ( S ) ) <- [89

for all S E ~}.

(3.10)

Now (3.10) contains many redundant constraints, in fact enough that the dual problem always has an integer solution, provided c is integer valued. Theorem 3.4. If c is integer valued, then there exists an integer optimal dual solution to the linear program: maximize cx subject to (3.4), (3.5) and (3.10).

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Proof. By Theorem 3.2 there exists a discrete optimal dual solution y to the problem of maximizing cx subject to (3.4), (3.5) and (3.6). Let X be the set of nodes i for which yi is fractional. By linear programming duality

~, biYi + ~, qsYs = cx

iEV

SE~

for an optimal b-matching x and since c is integer valued, it follows that ~,iex biyi is integer valued, thus b ( X ) must be even. If we lower yi by 89for every i ~ X, let yx = 1 and let Ys = 0 for all other S ~ ~ - ~, we obtain an integral optimal solution to the dual problem of maximizing cx subject to (3.4), (3.5) and (3.10). Note that in order to obtain an integer optimal dual solution, we let Yx be positive for a set X E ~ - ~. Fig. 1 is an example of a b-matching problem which has an integer optimal dual solution only if we are allowed to use such a dual variable. We obtain a maximum matching with value 3 by letting xj -- 1 for each edge j. Since b(S) is even for every S _C V, ~ = ~ and ~,i~v biYi is even for any integer dual solution y. Thus if we are restricted to dual variables corresponding to sets S in ~ and nodes in V we cannot obtain an integer valued dual solution whose objective value is less than 4. The problem that we now study is: When does the dual problem (3.7) (i.e., when we are restricted to ~ instead of ~ ) , have an integer optimal solution.

4. Bicritical graphs In this section we introduce the concept of bicritical graphs and establish some of their basic properties. In the following sections we show the role that these graphs play in characterizing b-matching problems that have an integer optimal dual solution. The bicritical graphs defined in this section are a generalization of the bicritical graphs introduced by Lovfisz [5] and Lovfisz and Plummer [6] who were concerned with obtaining bounds on the number of perfect b-matchings possessed by a graph when bi = 1 for all i E V. Let b = (bi: i E V) be a vector of integers satisfying bi ->2 for all i E V. We call b' a simple reduction of b at v if 2

Fig. 1. b-matching problem requiring fractional optimal dual solution. All edge costs = 1.

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, [bi, b~=[bo-2,

iE V-{v}, i=v.

Thus we obtain a simple reduction of b by simply subtracting 2 from the degree constraint of one node. We say that G is weakly b-bicritical if bi -> 2

for all i E V,

(4.1)

for any simple reduction b' of b, there exists a perfect b'matching. (4.2) It is an immediate consequence of (4.2) that if G is a weakly b-bicritical graph, then b(V) is even.

(4.3)

Fig. 2 is an example of a weakly b-bicritical graph. We call a graph nontrivial if IVI -> 3. It is easily seen that the only trivial weakly b-bicritical graph consists of a single node i and bi = 2. Lemma 4.1. A nontrivial weakly b-bicritical graph has a perfect b-matching.

Proof. Suppose G is weakly b-bicritical and has no perfect b-matching. Let y* be an optimal dual solution to the b-matching problem, taking c; = 1 for all ] E E. Then its value is 21b(V) - 1. For any simple reduction b' of b at i, the perfect b'-matching is an optimal solution to the original problem so by complementary slackness (Theorem 3.3) we must have y~* = 0 for all i E V. For any S E ~ there is an optimum matching x having a deficiency of 2 at any node i ~ S, so x(3,(S))< qs and hence by Theorem 3.3 y$ = 0. Hence the dual solution has value 0 and b ( V ) = 2. This together with (4.1) shows that G is a trivial weakly b-bicritical graph. It should be observed that weakly b-bicritical graphs need not be connected; any union of disjoint nontrivial weakly b-bicritical graphs is itself weakly b-bicritical. We now prove an analogue of a theorem of Lovfisz and Plummer [6] which characterizes the nontrivial weakly b-bicritical graphs. First we introduce some 2

2 Fig. 2. A weakly b-bicritical graph.

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terminology. Let G = (V, E, ~) be a graph and let X _C V. We partition V - X as follows ~~

= {i E V - X: G[{i}] is a component of G[ V - X]},

~I(X) = {S C_ V - X : b(S) is odd, ISI->2 and G[S] is a component of G[ V - X]}, ~2(X) = {S_C V - X : b(S) is even, Isl->2 and G[S] is a component of G[ V - X]}. See Fig. 3. Tutte [9] proved the following characterization of those graphs possessing a perfect b-matching (see also Berge [2]). Theorem 4.2. G has a perfect b-matching if and only if for every X C_ V

b(X) >- b(C~~

+ Ic~'(x)l.

Now we prove a characterization of weakly b-critical graphs. Theorem 4.3. Let G be a graph with no trivial components. Then G is weakly b-bicritical if and only if b(K) is even for the nodeset K of each component and for every nonempty X C_ V,

b(X) -> b (cr176

+ I ~ ( X ) l + 2.

(4.4)

Proof. First we prove the sufficiency. If there exists i E V such that bi = 1, then setting X = {i} we would obtain a contradiction to (4.4) so bi -> 2 for all i ~ V. Now suppose that for some i ~ V, there is a simple reduction b' of b at i for which there does not exist a perfect b'-matching. Then by Theorem 4.2 there exists X C_ V such that

b'(X) < b'(~~

+ Ic~(X)I.

Fig. 3. Sample cr176 ~'(X), ~2(X). The integer beside each node i is the value of bl.

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If X = O, then since G has no trivial components and since b ( K ) is even for the nodeset K of each component we must have c~~ = ~I(X) = O. Thus b'(X) < O, a contradiction. Hence X # O. If i ~ V - X , then

b ( X ) = b ' ( X ) < b(CC~

+ Ice'(X) I

so (4.4) is violated for X. If i E X, then

b ( X ) = b ' ( X ) + 2 < b(CC~

+ Ic~'(X)l + 2

so (4.4) is violated. Thus the sufficiency is proved. To prove the necessity, let X be a nonempty subset of V. Let b' be a simple reduction of b at i E X. Since G has a perfect b'-matching, by Theorem 4.2

b ' ( X ) >- b'(C~~

+ Ic~'(X)l.

Therefore

b ( X ) = b ' ( X ) + 2 _> b(C~~

+ Ic~(X)[ + 2.

It follows from (4.3) that b ( K ) is even for the nodeset K of each component which completes the proof. It is interesting to compare Theorem 4.3 to the following characterization of b-critical graphs.

T h e o r e m 4.4 (Pulleyblank [7]). A connected graph G is b-critical if and only if

b ( V ) is odd and [or every nonempty X C_ V, b ( X ) >- b(C~~

+ I~I(X)I + 1.

Let b =(hi: i E V) be a vector of positive integers. We call an integer b ' = (b~: i E V ) a complex reduction of b if O<-b'<-b and b'(V) = b ( V ) - 2 . In other words, a complex reduction of b is obtained by either subtracting 2 from the value of bi for some i ~ V such that bi -> 2 or else by subtracting 1 from the values of bu, by for distinct nodes u, v E V. We say that G is strongly b-bicritical if for any complex reduction b' of b, G has a perfect b'-matching. There are essentially two trivial strongly b-bicritical graphs. The first consists of a single node i with bi = 2 (the trivial weakly b-bicritical graph). The second consists of two nodes u and v joined by any number of edges (including no edges) and for which bu = by = 1. The following is immediate. Strongly b-bicritical graphs have for any edge j a perfect bmatching x such that x~ > 0. (4.5) The following characterizes nontrivial strongly b-bicritical graphs.

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Theorem 4.5. G is a nontrivial strongly b-bicritical graph if and only if G is connected

and

G has no cutnode i

IV[ -> 3,

for which bi = 1,

b ( V ) is even, b ( X ) >- b(%'~

(4.6) (4.7) (4.8)

+ Ics

+ 2,

(4.9)

for any X C_ V such that b ( X ) >_2. ProoL The necessity of (4.6)-(4.8) is easily seen and the necessity of (4.9) can be shown in a manner analogous to that used in the proof of Theorem 4.3. We now prove the sufficiency. Suppose G satisfies (4.6)-(4.8) but is not a nontrivial strongly b-bicritical graph. Then there exists a complex reduction b' of b such that G has no perfect b'-matching. Therefore by Theorem 4.2 there exists X _C V such that

b'(X) < b'(~g~

+ I~g"(X)[

where ~'l(X) is the set of node sets S of components G[S] of G [ V - X ] for which b'(S) is odd and IsI > 1. If b ( X ) = 0, then X = 0 and so G is not connected, contrary to (4.6). Suppose b ( X ) = I. If G [ V - X ] has two or more components, then X consists of a cutnode i for which b; = 1, contradicting (4.7). If G[ V - X] is connected, then by (4.8), b'(X) and b'(CC~ + I~g'l(X)l must have the same parity and hence differ in value by at least two. Therefore G [ V - X] consists of a single node u with bu -> 3. But then I VI = 2, contradicting (4.6). Therefore b ( X ) -> 2. Since

( b ( X ) - b'(X)) + (Icg't(X)l -I
b ( X ) < b'( ~~

+ lqgt(X)f + 2

<--b(~~

+ I~I(X)I + 2

so (4.9) is violated and the proof is complete. Theorem 4.6. A connected weakly b-bicritical graph is strongly b-bicritical.

Proof. If G is trivial, then so too is the result. Otherwise (4.6)-(4.8) are immediate and (4.9) follows from Theorem 4.3. Thus the result follows from Theorem 4.5. In view of Theorem 4.6 we simplify our terminology and say that G is b-bicritical if it is a connected nontrivial weakly b-bicritical graph. Edmonds [4] introduced the concept of a "good" characterization. A charac-

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terization is called good if it can be checked in time which grows polynomially with the size of the problem. Since we can determine in polynomial time whether or not a graph has a perfect b-matching, at most IV[ applications of this algorithm will determine whether or not a graph is b-bicritical. Thus there exists a priori a good characterization of which graphs are b-bicritical (connectivity and nontriviality being easy to check). However, Theorem 4.3 gives a good characterization of weakly b-bicritical graphs (and hence of b-bicritical graphs) independent of the b-matching algorithm or Theorem 4.2. Thus we can show a graph G is weakly b-bicritical by exhibiting [VI b-matchings, one for each simple reduction of b. We can show that G is not weakly b-bicritical by exhibiting a single nonempty X C_ V which violates (4.4). 5. Dual integrality with unit edge costs We now turn to the main subject of this paper, determining when we require a fractional optimal dual solution if all edge costs are equal to 1. First we observe that in this case, Theorem 3.2 specializes to the following. Theorem 5.1. If cj = 1 for all j E 17, then there exists an optimal dual solution y* satisfying y~ ~{0, 1} f o r a l l S E ~ ,

(5.1)

y~'~{0,2l, 1} f o r a l l i ~ V.

(5.2)

In fact when all edges costs are equal to 1, optimal discrete dual solutions have a particularly simple structure, as described in the following lemmas. Lemma 5.2. L e t y* be an optimal discrete dual solution. Then for any S E ~ such that y~ = 1 there exists at most one i E S such that y~' > 0 and if such an i exists, then bi = 1. Proof. Let X = {i E S: y~* = 1}, Y = {i ~ S: y* = 89 We define a vector y' by y~=

Y~'=

1 if i E X , 21 if i E S - X , y* i f i E V - S ,

{o

,f

-s

if T E n - { S } .

Then y' is easily seen to be a discrete dual solution. Moreover its value exceeds that of y* by ~b(S-(X

U Y)) - 89

1) = ~(1 - b ( X ) - b(Y)).

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Since y* is optimal, this value is nonnegative. Hence b ( X ) + b( Y) <- 1. Since bi -> 1 for all i, the result now follows. Lemma 5.3. Let y* be an optimal discrete dual solution. Then for any S, T ~ such that y~ = y~ = 1 we have b(S O T) <- 2. Proof. We define a feasible dual solution y to the problem (3.4), (3.5) and (3.10) by letting Yi = Y,* for all i E V, and

Yx=

t

X = S o r T, if X = S U T , if x ~ - { s , T, SU T}, if X E ~ - ( . ~ U{SU T}).

The value of y exceeds that of y* by

[b(S~_T,J_(b(S~-,.)_(b~-l) -b(SO_2T'+2 This value must be nonnegative so b(S n T)<_ 2. Fig. 4 gives examples to show that the bounds given in Lemmas 5.2 and 5.3 can in fact be achieved. If G is b-bicritical, then an optimal discrete dual solution is uniquely determined. The only optimal y* is obtained by letting y~' = l2 for all i ~ V and letting y~ = 0 for all S ~ ~. To prove this we first establish the following lemma.

Lemma5.4. Let G be weakly b-bicritical. Then for any optimal dual solution y*, we have y.*, <-I for all i ~ V. Proof. If G is trivial, then the zero vector is the unique optimal dual solution, so assume G is nontrivial. Suppose for some v E V, y* > ~. Since G has a perfect b-matching (by Lemma 4.1) the value of the dual solution y* is 89 If b' is a simple reduction of b at v, then we obtain a feasible dual solution y to the problem (3.4), (3.5), (3.10) with b' by extending the domain of y* to ~ by letting /

"%

\~, YS = I ~ - ~

dge j with x I = 1 --

edge j with xj =0

Fig. 4. Sample optimal dual solutions.

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Ys = 0 for all S ~ ~ - ~ . Its values is ~ b ( V ) - 2 y * which is less than ~ b ( V ) - 1 . But since G is b-bicritical, there exists a solution to (3.4), (3.5), (3.10) with b' having value 89 1. Thus we have contradicted linear programming duality. We now give a sufficient condition for a graph to require a fractional dual solution. Theorem 5.5. If G is a b-bicritical graph, then the only optimal discrete dual solution y* is obtained by letting yi* = 89for all i E V and y$ = 0 for all S ~ ~. Proof. It will be sufficient to show y$ = 0

for all S E ~

(5.3)

for since G has a perfect b-matching (by L e m m a 4.1), by linear programming duality the value of y* is ~b(lO. Thus L e m m a 5.4 and (5.3) imply y; = 12for all iEV. Let ~ + = { S ~ . ~ : y ~ = 1} and suppose ~ + ~ 0 . (Note that by Theorem 5.1, y* = 0 for all S ~ ~ - ~+.) Since bi -> 2 for all i E V, we have by Lemma 5.2 that y~' = 0

for all i ~ S, for all S E ~+.

(5.4)

Let j E ~(S) for some S E ~ + . If there is no T E ~ + such that j E y(T), then y*(~(j))+ y * ( ~ ( j ) ) = y*(O(j))-<~ by L e m m a 5.4 and (5.4). But this contradicts the feasibility of y* so for any S E ~t +, for any j ~ 8(S) that j ~ y(T).

there exists T ~ ~+ such

(5.5)

By Lemma 5.3 we have for a n y S , T E ~ + ,

b(SAT)~2

and

[ S A T I_<1.

(5.6)

Let S E ~ +. Then since b(S) is odd but b(V) is even and G is connected, by (5.5) and (5.6) there is v E S and T ~ ~+ - {S} such that {v} -- S n T and by = 2. If v is the only member of S belonging to a member of ~ + - { S } then by (5.5) v is a cutnode of G. Hence if we reduce bi by 2, G [ S - {v}] is "disconnected" from the rest of G. But b ( S - {v}) is odd, so no perfect matching can exist, contradictory to G being b-bicritical. Hence there exists u ~ S - {v} and U E ~+ - {S, T} such that b, = 2 and {u} = S n U. See Fig. 5. Now let x be a perfect matching of G. By complementary slackness (Theorem 3.3) x is a np-matching of G[T] so there is some j E y(T) n 8(v) for which xj -> 1. Similarly x is a np-matching of G[U] so there is some k E 8(u) n y ( U ) for which xj > 1. But {j, k} C ~(S) so x(8(S)) - 2 and hence x(y(S)) <_89 - 2). Thus x cannot be near perfect on G[S] which contradicts Theorem 3.3 for S since we assumed x and y* to be optimal. Therefore, ~+ -- 0 and so (5.3) is established and the result follows.

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189

S

t

/

x

\

Fig. 5.

For any feasible dual solution y, we let to(G; y, b) denote the value of this dual solution relative to this dual objective function. That is, to(G; y, b) = ~'~ (y,b: i E V) + ~, (ysqs: S E ~). (Since qs = ~ ( b ( S ) - 1) we do not include q as a parameter.) If we omit the first parameter, then the graph G is understood, i.e., to(y, b) -~ to(G; y, b). We now establish some structural properties of graphs that do not possess an integer optimal dual solution. We frequently wish to " c o m b i n e " dual solutions in the following fashion. Let y be an optimal discrete dual solution and let X = {i E V: Yi = 21}. Let y be any feasible dual solution to the b-matching problem restricted to G[X]. The modification of y by y is defined to be the vector 9 defined by ~. = {y, Yi

ifiEX, ifiEV-X,

j'Ys i f S ~ Ys i f S ~

and and

S_CX, S~X.

Proposition 5.6. The modification of y by y is dual feasible. Proof. We must show that 9($(j)) + 3~(~(j)) -> 1 for all j E E. This is obvious for j E ~/(X) U 3,(V - X). Let j E 8(X). Then 9($(J)) + 9(~(J)) -> Y($(J)) + Y(~(J)) - 12. But since exactly one end of j had a fractional y, we must have had y($(j))+ y(~(j)) > 189 since y was dual feasible. The result now follows. By L e m m a 5.2, if S E ~ immediate.

and SC_X, then y s = 0 . The following is now

Proposition 5.7. o2(9, b) = co(y, b) - ~b(X) + r

y, b).

As a result of Proposition 5.7 we have the following "local optimality" condition.

190

W. Pulleyblankl Dual integrality in b-matching problems

Lemma 5.8. Let y be an optimal discrete dual solution to a b-matching problem and let Z = {i E V: Yi = 12}.Let G[H] be any component of G[Z]. If we let y be the restriction of y to H U {S E ~: S C_H}, then y is an optimal dual solution to the b-matching problem on G[H]. Moreover, for any other optimal dual solution ~ to the b-matching problem on G[H], the modification of y by ~ is an optimal dual solution to the b-matching problem on G.

Proof. Clearly y is a feasible dual solution to the b-matching problem on G[H], and the modification of y by y is of course just y. Now let ~ be any feasible dual solution to the b-matching problem on G[H]. All we need show is that if we let 33 be the modification of y by ~, then 33 is a feasible dual solution to the original problem and the result will follow from Proposition 5.7. Let ] E E. If ] E T(H) or j E y ( V - H ) , then clearly 33($(])) + 33(~(])) -> 1. If j E 8(H), then since H is the node set of a component of G[Z], if {v} = $(j) - H, then v ~ V - Z . Then as we saw when proving Proposition 5.6, y(~b(j))+ y(~(])) _> l89 and so 33(~/,(]))+ 33(~(])) > 33v+ 9(~(J)) -> 1. We now give necessary conditions for a b-matching problem to have no integer optimal dual solution. Theorem 5.9. Let G and b be such that there exists no integer optimal dual solution. Then there exists X C_ V such that bi >- 2

for all i ~ X,

[or every optimal matching x,

(5.7) x(8(i)) -- bi, for all i ~ X,

at least one component of G[ V - X] is b-bicritical, if Z is the set of nodes in the b-bicritical components of G[ V - X ] , then [or every optimal matching x, x~ = 0 for all ] ~ 8(z) u ~(x), there is an optimal discrete dual solution y* which is fractional for precisely those nodes in Z.

(5.8) (5.9) (5.10) (5.11)

See Fig. 6. Proof. Let y* be an optimal discrete dual solution for which Z = {i E V: y* = 2l} is minimal. Since there exists no integer optimal dual solution, Z # f l . Let X = N(Z). Clearly (5.11) is satisfied; we show that (5.7)-(5.10) are also satisfied for X and Z so defined. Let G[H] be a connected component of G[Z]. We wish to show that G[H] is b-bicritical. By L e m m a 5.8, the restriction of y* to H U { S E ~ : S C_H} is an optimal dual solution to the b-matching problem on G[H]. The value of this solution, ~b(H) must be integral and hence b(H) must be even.

W. Pulleyblank/ Dual integrality in b-matching problems

191

x(8(i)): b i for every

xj=O for every optimal matching Fig. 6. Structure when there exists no integer optimal dual solution.

Suppose G[H] is not b-bicritical. If G[H] is trivial, then let v be the single node in G[H] for which by = 2. Since y* is optimal and y* = 89 we must have 8(v) ~ 0. Let j E 6(v), where ~(j) = {u, v}. Since u E Z we have Y*(CK]))+ y*(~(j))-> 189 Thus we can reduce y* to zero, preserving feasibility, contradicting optimality of y*. Hence G[H] is nontrivial, so by Theorem 4.3 there exists nonempty M C H such that

b(M) <

b(~~

+

Ic~l(M)l + 2,

where ~0, cr and ~2 are taken relative to G[H]. We define a dual solution ] to the b-matching problem on G[H] (using ~ instead of ~) by

{i

)~i= I {1 Ys =

ifiEM'

ifi~SE~2(M), if i E H - (M U {S: S E ~g2(M)}), if S E cd'(M), if S E ~ - ~l(M).

It is easily checked that ~ is dual feasible. Moreover to(G[H]; ~, b) =

b(M)+ 89~

< 89b(M) + 89176

(b(M):

S E ~2(M))+ 2~, (89

+89~g1(M)] +

1): S E ~ ' ( M ) )

1 - 89~r

+ 89~ (b(S): S E ~ ( M ) U cg2(M)) =

89

+ 1.

Thus to(G[H]; ~, b) <-89 = to(G[H]; y*, b). Moreover, {i E H : Yi = 89 C H. It may happen that ]s = 1 for some S E ~ - ~, in this case we can simply find an optimal discrete dual solution y' to the b-matching problem on G[S] and make y conform with y' on S O ~[S]. Thus we are able to obtain an optimal discrete dual solution to the b-matching problem on G[H] having a smaller set of fractional components than y*. Hence if we modify y* by this solution, we obtain an optimal discrete dual solution which contradicts the minimality of Z. Hence G[H] is b-bicritical. If some component G[H] of G[V-(XUZ)] were b-bicritical, then by

192

W. PuUeyblank/ Dual integrality in b-matching problems

Theorem 5.5 the unique optimal discrete dual solution to the b-matching problem on G[H] is obtained by defining y~ = 89for all i E H. We could then perform the obvious modification to y* which would contradict optimality of y*. (Since y * = I for all i E X , and since j E S ( H ) implies j ~ 8 ( X ) we know this modification will be dual feasible.) Thus G[Z] consists precisely of the b-bicritical components of G[ V - X], and at least one such component exists, establishing (5.9). By L e m m a 5.2 there can be no S E 9 such that S n Z ~ I~ and y~ = 1, for since every component of G[Z] is b-bicritical, we must have bi - 2 for all i ~ Z. Therefore, for y* to be feasible, we must have y* = 1 for all i E X. Therefore, (5.8) follows from complementary slackness Theorem 3.3. Moreover,

y*(t~(j)) + y*(~(j)) > 1 for all j E 8(Z) U y(X) so again by complementary slackness, xj = 0 for any such j in any optimal matching, establishing (5.10). Finally suppose b~ = 1 for some i E X. Let H be the set of nodes in components of G[Z] which contain a node adjacent to i. We show that G[H U {i}] is b-critical. That is we show that there is a np-matching x of G[H U {i}] deficient at any v E H O {i}. This follows from L e m m a 4.1 for v = i, since G[H] is nontrivial and weakly b-bicritical. For v ~ N(i) we can find a b-matching x of G[H] which is perfect for u E H - { v } and which has a deficiency of 2 at v (since G is weakly b-bicritical). Then by setting xj = 1 for the edge j joining v and i we get the desired result. Finally for v ~ H - N(i), let u ~ N(i) belong to the same component of G[Z] as v. Construct a b-matching x perfect for H - {u, v} and having a deficiency of 1 at u, v (which is possible by Theorem 4.6). Set xi = 1 for the edge ] joining u and i and we get the desired result. Hence G[H U{/}] is b-critical. Thus we c a n modify y* by letting y* = 0 for v E H U {i} and ySu{i}= 1 and obtain a discrete dual solution whose value is no greater but which has fewer fractional components, contradicting our choice of y*. Hence b~ -> 2 for all i E X, (5.7) is established and the proof is complete. Unfortunately if not all the bi are even, conditions (5.7)-(5.11) are not sufficient to require a fractional optimal dual solution, as can be seen from Fig. 7.

a

Fig. 7. Example of insufficiency of (5.7)-(5.11).

w. PulleyblanklDual integrality in b-matching problems

193

Although in this case an optimal dual solution y* is obtained by letting y~* = 21for i E Z , y ~ = l for i E X and y ~ ' = 0 for i E V - ( Z U X ) , w e can in fact find an integer optimal dual solution y by letting Ya = 1, Ys = 1, and all other components of y be 0. In general, the problem is determining when a b-bicritical graph is contained in a b-critical graph, whose dual variable can be used to enable us to avoid making the nodal dual variables in the b-bicritical part become fractional. Thus a question which is still open is that of finding necessary and sufficient conditions for a graph to require a fractional optimal dual solution when all edge costs are 1. However, if bi is even for all i ~ V, then we obtain the following converse results.

Theorem 5.10. Suppose bi is even [or all i E V and let Z C V be such that G[Z] is b-bicritical and x i = 0 for every ] E 8(Z), for every maximum matching x. Then for any optimal dual solution y* we have yi* = ~ .for all i E Z. Proof. Any maximum b-matching of G[Z] can be extended to a maximum b-matching of G, since every maximum matching x of G has xj = 0 for all ] E 8(Z). Since G[Z] is b-bicritical, by (4.5) and Theorem 4.6 for each edge j E 3/(Z) there exists a perfect b-matching x of G[Z] such that xj > 0. Therefore, by complementary slackness (Theorem 3.3) y*(~b(j)) = 1 for all j E y(Z). (Since bi is even for all i E V, .~ = ~.) Thus the restriction of y* to Z is a solution to this system of equations, but since G[Z], being b-bicritical is connected and nonbipartite, we have y~* = ~ for all i E Z. Thus, when b is even, we have the following characterization of optimal dual solutions with a minimal set of fractional components.

Theorem 5.11. Suppose bi is even for all i ~ V and let y* be an optimal discrete dual solution such that the subgraph of G induced by Z = {i E V: y~' = ~} is nontrivial and weakly b-bicritical. Then y* has a minimal set of fractional components. Proof. All we need show is that xj = 0 for every j E 8(Z) for every maximum b-matching x and the result follows from an application of Theorem 5.10 to each component of G[Z]. Let j E 8(Z) and let {u, v} = ~b(j) where u ~ Z, v E V - Z. Then y* = 21and since y* is feasible and y* is integer valued, we must have y* = 1. But then y*(~b(j)) > 1 so by complementary slackness Theorem 3.3 xj = 0 for every optimum b-matching x. We close this section by noting the following corollaries of Theorem 5.9.

Corollary 5.12. If G is a minimal graph which does not have an integer optimal dual solution, then G is b-bicritical.

194

W. PulleyblanklDual integrality in b-matchingproblems

Corollary 5.13. If bi = 1 for all i E V, then there exists an integer optimal dual solution.

6. Arbitrary integer edge costs Now we give some results concerning b-matching problems in which the edge costs are arbitrary integers. First we note that for any edge j such that c~ -< 0, we can assume xj = 0 in any optimal matching x. Thus we can ignore all such edges. Moreover, any feasible dual solution to the problem obtained by deleting these edges will also be a feasible dual solution to the problem obtained with them present. Hence we assume that all edge costs are positive integers.

Proposition 6.1. If cj is even for all j ~ E, then there exists an integer optimal dual solution.

Proof. Let y* be an optimal discrete dual solution to the problem obtained by replacing c~ with 89 for all j E E. Then 2y* is an integer optimal solution to the original problem.

We can also obtain the following analogue of Theorem 5.9.

Theorem 6.2. Let G, b and c be such that there does not exist an integer optimal dual solution. Then there exists Z C_ V such that every component of G[Z] is b-bicritical and there exists an optimal dual solution y* which is fractional for precisely the nodes in Z.

Proof. Let y* be an optimal discrete dual solution for which Z = {i E V: y,* is fractional} is minimal. Then Z r ~. We show that every component of G[Z] is b-bicritical. Let y = [y*]. That is, we truncate the fractional members of y*. Then y is not a feasible dual solution, but for any edge j such that y(~b(j))+ y(~(j)) < ci we will have j E y(Z) and y(~b(j)) + y(~(j)) = cj - 1. Thus if we add to y any feasible dual solution y' for the b-matching problem on G[Z] taking all edge costs to be 1, we will obtain a feasible dual solution )~ to the original problem. Moreover the only nodes i for which 9~ will be fractional are those nodes i in Z for which y} is fractional. Since Z is minimal, the only discrete optimal dual solution y' to the problem on G[Z] is obtained by letting y~ = ~ for all i E Z and y~ = 0 for all S. Therefore it follows from Theorem 5.9 that G[Z] consists of bobicritical components.

W. Pulleyblank/ Dual integrality in b-matching problems

195

Corollary 6.3. If bi = 1 .for all i E V, then .for any integer objective function c there exists an integer optimal dual solution. (This has also been proved by Cunningham and Marsh [3].)

7. Restriction to facet dual variables

We mentioned in Section 3 that there may exist i ~ V such that x(8(i)) <- bi is not a facet of the matching polytope and S C 2 such that x(y(S))-< qs is not a facet of this polytope. We conclude by showing that the corresponding superfluous dual variables we have used do not affect the problem of finding an integer optimal dual solution. Let 2 ' = { S ~ 2 : x ( y ( S ) ) < - q s is a facet} and V' = {i E V: x(3(i)) <- bi is a facet}. All we need show is that if we can obtain an integer optimal dual solution using dual variables corresponding to members of V and 2, then we can also find an integer optimal dual solution using dual variables corresponding to members of V' and 2'. The following characterizes 2 p.

Theorem 7.1 (Pulleyblank [7]). 2 ' = { S E 2 :

G[S] contains no cutnode v for

which b~ = 1}. Thus for any $ E 2 - 2 ' , G[S] decomposes into " w e a k " blocks, i.e., S = S ~ U S z U ' " U S k where for each i ~ { 1 , 2 ..... k}, S i E 2 ' , if S~ASjgfJ, then b(Si O Sj) = IS~ n Sj[ = 1 and U~=l y(Si)is a partition of y(S). (See Pulleyblank [7] for details.) Hence if an optimal dual solution y* had y~ = r we could instead have set y~, = ,e for i E {1, 2 ..... k} and y~ = 0. This is easily seen to be feasible and a straightforward induction on k shows that k

b(S)

- 1 = Y~ ( b ( S , ) - 1) i=1

so this modification will retain the same objective value. Now we analyze nodal dual variables. A balanced edge is a graph consisting of two nodes u, v joined by one or more edges and satisfying b, = b,. The following characterizes the essential nodal dual variables. Theorem 7.2 (Pulleyblank [7]). V - V' = {i ~ V:

b ( N ( i ) ) < bi and if b ( N ( i ) ) = bi, then i belongs to a component o.f G which is not a balanced edge

(7.1)

b ( N ( i ) ) = bi + 1 and y(N(i)) ~ 0}.

(7.2)

or

See Fig. 8.

196

W. Puileyblank/ Dual integrality in b-matching problems 7

3 3

2

5

Nodes not corresponding to primal facets: 9

Fig. 8. Inessential nodal dual variables.

Now suppose i E V - V ' and y~*= e > 0 for an optimal dual solution y*. If (7.1) applies, we simply set y~' to 0 and add e to the dual variables of the nodes in N(i). If (7.2) applies, we set y~* to 0 and add ~e to the dual variable y} where S = N ( i ) U {i}. (It is easily checked that G [ N ( i ) U {i}] is b-critical and has no cutnode v for which by = 1 and hence S E .9.'.)

Acknowledgements The bulk of this work was carried out while the author was a research fellow at CORE, University Catholique de Louvain Belgium. I am indebted to G6rard Cornu6jols for several helpful discussions during the preparation of this paper as well as Martin Gr6tschel and an anonymous referree for careful readings of an earlier version and suggested improvements. This work was supported in part by the National Research Council of Canada.

References [1] A. Aho, J. Hopcroft and J. Ullman, The design and analysis of computer algorithms (AddisonWesley, Reading, MA, 1974). [2] C. Berge, "Sur le couplage maximum d'un graphe", Comptes Rendus Hebdomadaires des s~ances de I'Acad~mie des Sciences, 247 (1958) 258-259. [3] W.H. Cunningham and A.B. Marsh III, "A primal algorithm for optimum matching", Mathematical Programming Study 8 0978) 50-72. [4] J. Edmonds, "Paths trees and flowers", Canadian Journal of Mathematics 17 (1965) 449-469. [5] L. Lov~isz, "On the structure of factorizable graphs", Acta Mathematica Academiae Scientiarum Hungaricae 23 (1972) 179-195. [6] L. Lov~isz and M.D. Plummer, "On bicritical graphs", Colloquia Mathematica Societatis Jtinos Bolyai 10 (1973) 1051-1079. [7] W. Pulleyblank, "The faces of matching polyhedra", Thesis, University of Waterloo (Waterloo, Ont., 1974). [8] W. Pulleyblank, "Minimum node covers and 2-bicritical graphs", Mathematical Programming, 17 (1979) 91-103. [9] W.T. Tutte, "The factors of graphs", Canadian Journal of Mathematics (1952) 315-318.

Mathematical Programming Study 12 (1980) 197-205. North-Holland Publishing Company

A TECHNIQUE FOR DETERMINING BLOCKING ANTI-BLOCKING POLYHEDRAL DESCRIPTIONS

AND

H.-C. H U A N G

Nanyang University, Republic o[ Singapore L.E. T R O T T E R , Jr.

Cornell University, Ithaca, NY, U.S.A. Received 24 October 1977 Revised manuscript received 7 March 1978 A general method is described for determining blocking and anti-blocking polyhedra related to any combinatorial family given as the extreme points of a polyhedron. This technique is illustrated in detail for the common independent sets of two matroids.

Key words: Blocking Polyhedra, Anti-Blocking Polyhedra, Matroids, Matroid Intersection.

1. Introduction Suppose M is a nonnegative rational matrix with n columns and let ~ = {x ~ R~: Mx >-1}, where R~ denotes the nonnegative orthant of R" and 1 is a vector of appropriate dimension, each of whose components is one. Fulkerson has shown in [6, 8] that associated with each such p o l y h e d r o n ~ is a unique blocking polyhedron ~ = {x E R~: ~7=~ yjxj _> 1, Vy E ~}. Furthermore, ~ is of the same form as ~ ; i.e., ~ = {x U R~-: Bx > 1} for a nonnegative matrix B. Also = ~. Any such nonnegative matrix B is called a blocking matrix for M. (Observe that we do not require the elimination of inessential rows for matrices M and B; i.e., M and B need not be " p r o p e r " , as in [6].) Analogously, when M has no zero columns, so that M = {x E R~: Mx <-1} is bounded, Fulkerson [7, 8] associates with M the unique anti-blocking polyhedron s~ = {x E R~-: ~7=1 yjxj --< 1, Vy E M}. H e r e too, ~ = M and s~ = {x E R~: Ax <- 1} for a nonnegative matrix A with no zero columns, called an anti-blocking matrix for M. Thus the blocking relation pairs polyhedra of the form ~ above, whereas the anti-blocking relation pairs polyhedra of the form M; these relations provide respective frameworks for viewing the paired occurrence of certain combinatorial theorems. For blocking polyhedra in R2 such results concern a maximum rational packing of extreme points of ~ (or ~ ) into a vector w ~ R2; for anti-blocking polyhedra in R2 attention is directed to a minimum rational covering of w E R2 by extreme points of M (or s~). For specific examples the reader is referred to references [1, 2, 6-9, 11, 13-17]. Thus, given a polyhedron 197

198

H.-C. Huang and L.E. Trotter Jr. l Blocking and anti-blocking polyhedra

_CR~ whose extreme points correspond to some combinatorial family of interest, it is natural to attempt deducing rational packing and covering results for members of this family by determining a blocking polyhedron ~ for = {x G R~-: x-> y for some y E ~} and (when ~ is bounded) an anti-blocking polyhedron sr for ~ = {x E R~: x -< y for some y E ~}. In this paper we describe a general method for determining linear systems which define polyhedra ~] and when polyhedron ~ is given by any linear system. This technique is based on results which relate blocking and anti-blocking polyhedra to generalized circulations (see [ 14]) and on the algebraic operations of deletion, contraction and projection for manipulating blocking and anti-blocking polyhedra (see [8]). In the final section of the paper we illustrate this technique on a "matroid intersection" example.

2. Procedure Suppose ~ ={x ~ R g : Mx >-1} where M is a nonnegative matrix with n columns and let B be a blocking matrix for M so that ~ = {x E R~: Bx >- 1} is the blocking polyhedron of ~. When M has no zero columns, we define sr = {xER~-: Mx<_ 1} and let A be an anti-blocking matrix for M with ~ = {x E R~: Ax <- 1} the anti-blocking polyhedron of M. By contraction of the jth column of M is meant the removal of column j from M; deletion of column j is the removal of this column along with all rows which have a positive entry in column j. In polyhedral terms, contraction of the jth column of M corresponds to intersecting ~ with the hyperplane ~ = {x E R": xi = 0}, whereas deletion of column j corresponds to projecting ~ onto ~. These operations are used on blocking pairs of polyhedra as described in the theorem below. Since for an anti-blocking polyhedron M, 0 -< y -< x E M implies y E M, the projection of M onto is identical tO its intersection with ~. Thus for anti-blocking matrices we define the single matrix operation, projection of column j, as the removal of column j. Theorem 1 (Fulkerson [6, 7]). Suppose matrices M, B, A and polyhedra ~, ~, M, ~l are as described above. Also let matrices M', B', A' be obtained from M, B, A by respective contraction (projection), deletion and projection of column j, and define the resulting polyhedra : ~ ' = {x E R~--I: M ' x >- 1},

~ ' = {x E//R~-1: B ' x >- 1},

..~' = {x E R~-l: M ' x <- 1},

,~' = {x E R~-l: A ' x <- 1}.

Then ~ ' and ~ ' are a blocking pair of polyhedra and sg' and ~ ' are an anti-blocking pair of polyhedra. Suppose N is a real matrix with n columns and suppose we are given n

H.-C. Huang and L.E. TrotterJr.~Blocking and anti-blocking polyhedra

199

intervals /j, i - ]--< n, where /i =/at, bi] with 0 - a t -< b~. We list as the rows of matrix M the extreme points of the polyhedron = {x ERn: Nx = 0 and xj~Ij, Vj}. Theorem 2 below, which specifies blocking and anti-blocking matrices for M, is proved in [14]. We recall that the support of the vector k E R n is the set {j: k~#0}; K+ ={j: kj>0} and K_ ={j: kj<0} denote respectively the positive and negative support of k. Also denote the row space of N by ~ and recall that nonzero vectors of ~ with minimal support are termed elementary vectors of ~. We denote by ~:(~) the collection of all elementary vectors of ~. Theorem 2. (i) The blocking polyhedron of ~ = {x E R~: Mx >- 1} is given by

{

xj>-ai,

jEJ

j = l ..... n,

k~xj> ~, (-kj)a i jEK_

~

jEK+\ J

(ii) I[ M has no zero columns, {x ~ R~-: Mx <- 1} is given by { 0---xj--- bj,

~kjxj<_ ~ ( - k j ) b j .iEJ

jEK-

kjbi, kE~;(~),JC_K+. the anti-blocking polyhedron of M =

j = 1..... n,

~,

kjaj, k ~ ( ~ ) , J C K + .

jEK+\ J

Notice that the description of the blocking polyhedron of g~ and the antiblocking polyhedron of M in Theorem 2 using elementary vectors of ~ actually requires only finitely many constraints, since there are only finitely many vectors of ~ ( ~ ) up to scalar multiplication. Also observe that we could broaden the stipulation in Theorem 2 that k E ~ ( ~ ) to any k E ~. This follows from the fact (see [5]) that any k E ~ is a conformai sum of vectors in ~:(~); i.e., k = k I +... + k", where for 1 - i - m, k; E ~ ( ~ ) , K~+_CK+ and K/_ C_K_. Theorm 2 provides a means for determining both blocking and anti-blocking results for a given matrix M - - w h e n M is the list of extreme points of a polyhedron ~ as described above. This situation is completely general in the following sense (see also [1]). Suppose we wish to determine blocking and anti-blocking results for the extreme points of an arbitrary bounded polyhedron ~ ' C R ~ , given as the solution set to a system of linear equalities and inequalities. Then it is routine to transform the representation of ~ ' into the form of ~ above by the addition of slack variables and appropriate coordinate bounds. (If ~ ' is unbounded in coordinate ], we take bj arbitrarily large in Theorem 2--see [14].) Furthermore, an elementary argument establishes a one-to-one correspondence between the extreme points of ~ ' and those of ~. That is, the matrix M', whose rows are the extreme points of ~', is a full-rowed submatrix o f M, whose rows are the extreme points of ~. Thus we may apply

200

H.-C Huang and L.E. Trotter Jr. [ Blocking and anti-blocking polyhedra

Theorem 2 to obtain a blocking matrix B for M and an anti-blocking matrix A for M. Since M' is obtained from M by contracting (projecting) certain of its columns, Theorem 1 implies that a blocking matrix B' for M' and an antiblocking matrix A' for M' may be obtained, respectively, by using the column deletion operation on B and by projecting certain columns of A. We next illustrate this process in detail with an example.

3. Example In [3] Edmonds has demonstrated that the incidence vectors of comnon independent sets of maximum cardinality for two matroids defined on the set E are the extreme points of the following polyhedron ~ ' C R~ f, z( S ) <- rl(S),

SC_E,

z( T) <---r2(T),

Tc_E,

z ( E ) = ro(E), zi>-O,

j~E,

(1)

where z(U), rI(U), r2(U) and ro(U) denote respectively, for U C E, ~j~v zi, the maximum size of a subset of U which is independent in the first matroid, similarly for the second matroid and the maximum size of a set contained in U which is independent in both matroids. Rewriting (1) in a form compatible with Theorem 2 gives the polyhedron ~ defined by (2) and (3) below. z(S) + Ss - rl(S)y = O,

S C_ E,

z(T)+tr-r2(T)y=O,

TCE,

z(E)

(2)

- ro(E)y = O.

0-< zj-< bj,

j~E,

O<-ss<-bs,

SCE,

O<-tr<--br,

TC_E,

(3)

l_
H.-C. Huang and L.E. Trotter Jr./Blocking and anti-blocking polyhedra

20I

jecture was shown true for the special case of transversal matroids by Weinberger [15, 17]. The general form of the conjecture was established independently by McDiarmid [13], Cunningham [2] and Edmonds and Giles [4]. The derivation which follows, based on the approach of Section 2, is due to Huang [11]. McDiarmid [13] also determined an anti-blocking matrix for M'; the anti-blocking analogue of the derivation below also yields this result. Let n = [E[ + 2 IEI+~+ 1 and let N denote the matrix with n columns whose rows are determined by the constraints (2), with ~ the row space of N. Also denote as in Theorem 2 = {x E R~-: M x >- 1},

~ ' = {z ~ R~I: M ' z >- 1},

and let ~ and ~ ' denote the respective blocking polyhedra for ~ and ~ ' . Theorem 2 indicates that ~ is described by lower bound constraints and elementary vector constraints. After deleting s, t, y coordinates, the lower bound constraints simply become nonnegativity restrictions on the z coordinates. Furthermore, in any constraint for ~ determined by k ~ ~ ( ~ ) , K_ must contain the coordinate corresponding to the variable y. Otherwise the right-hand side of this constraint is nonpositive, and hence the constraint is inessential (see [6]) for defining ~ and it need not be considered in determining ~ ' . It then follows from the fact that upper bound components may be taken arbitrarily large for all remaining coordinates that an elementary vector constraint is also inessential for defining ~ unless K+ ~ J = 0, i.e., unless J = K+. Finally, the deletion operation requires that we omit such a constraint in determining the blocking matrix for M' if it contains a positive coefficient for any coordinate corresponding to an s or t variable. Thus the general form of those elementary vector constraints defining ~ which are useful in determining ~ ' is

kjzj -> -k~,

(4)

where ky denotes the component of k corresponding to the variable y. In the following we assume given a vector k ~ ~:(~) which determines a constraint of the form (4) that is essential for defining ~ and we show that after deleting coordinates corresponding to s, t and y variables, this constraint is of a specific form (see (9) below) which defines the polyhedron ~ ' of interest. Since k ~ we may write k as a linear combination of the rows of N using multipliers -ors for S C E , --~'T for T C E and p for the final row of N. Suppose that the multipliers -O'sl, 1-< i- 0, 1 -< i ~ p, and ~'Ti > 0, 1 ~ j -- 0, since we have ky < 0. Thus we may assume that all multipliers O's~( 1 - i - p ) , rTj (1--] ~ q) and p are positive integers.

202

H.-C. Huang and L.E. Trotter Jr.I Blocking and anti-blocking polyhedra 2

-- orsp

incidence I vectors of I subsets of E]

TT I

T I

--o'sl

--

incidence vectors of I subsets of El

"rTq P

s

t

!

0

I

L

,...,

0

1

,I I -r,(S,) I I -r,(Sp)

I I I -r2(~r,) I I -r,(~r,)

I 0 I0

Fig. I. Row multipliers defining a constraint of the form (4).

Thus far we have used no " m a t r o i d " properties in the problem. We now do so by invoking the well-known fact that matroid rank functions are s u b m o d u l a r (see [3]), i.e., for rl we have rl(S) + r~(S') >- rl(S U S') + r~(S O S'),

S, S' C_E.

Suppose that for some distinct i and i', 1 - i, i' - p, we have oX _> trsr and both Si-~ Sr and Sr-~ S; are nonempty. Consider the vector k ' E ~ determined by altering the row multipliers which produced k as follows: increase the multipliers trs, usF and trs~ns~, both by the amount O'sr, reduce the multipliers O's~ and o% both by this same amount and leave the remaining multipliers unchanged. Now k' E so the comments following the statement of T h e o r e m 2 imply that the following constraint determined by k' is satisfied for each vector x -- (z, s, t, y) E ~ , k;zj - - k y ,

(4')

jEK~_

_

,

i

i.e., the constraint (4') is valid for ~. Furthermore, k j - k i for all j E K+ = K+ and it follows easily from the submodularity inequality that ky-> k'y. If this inequality holds strictly, i.e., if ky > k~, then (4) is inessential for defining ~ , a contradiction 9 H e n c e we must have that ky = k~, and we may replace k by k'. Note that in passing from k to k' the total g-weight, ~s_c~ trs, does not change. Thus after a finite number of applications of this process to both r and ~" multipliers we obtain a vector k ' E ~ determined by positive integer-valued multipliers crs~, rr~ and p for which the sets S~ and T) form two nested sequences: SI D_S;_ D_... D_ S'p,, T~ ~_ T~ ~ ... D_ T'q,. Furthermore, k' determines the same constraint (4) as k. N o w if k' ~ ~ ( ~ ) , then we may write k' as a conformal sum of elementary vectors of ~ , say k ' = k t + . . - + k =. Since this sum is conformal, the sets S~ and T; defining each k j must also be nested. Thus we may restrict consideration to those vectors k ~ ~:(~) for which the sets Si, 1 _ i ---p, and Tj, l<_j<_q, form two nested sequences: S, D_ S2 ~_ ... D_ Sp, T~_T2D_...D_Tq. Finally, by allowing copies of certain sets, we may also assume that O's, = 1, 1 --- i - p, and Trj = l, 1 - j ~ q; since the constraints of (2) for S = 0 and T = 0 are trivial, we also have that Sp ~ 0, Tq ~ 0.

H.-C. Huang and L.E. Trotter Jr. I Blocking and anti-blocking polyhedra

203

The nesting property of the sets Si and Tj is crucial to the remainder of our development. The importance of the structure obtained from two such nested families of sets is well-known in other problems concerning the intersection of two matroids (see [3, 10, 12]). Suppose p > p and consider the vector k ' E ~ which is the row of N corresponding to the set Sp. Since p > p, the support of k contains that of k', and because k E ~=(~), k' must be a nonzero scalar multiple of k. This is impossible, since vectors k and k' disagree in sign on z coordinates but agree in sign on the y coordinate. Thus we must have p -< p, and similarly, q -< p. Suppose q < p and consider the vector ! E ~ defined by subtracting the row of N corresponding to the set St from the last row of N. Since q < p, it is clear that the support of k contains that of l. Thus k must be a nonzero scalar multiple of I. But this implies q = 0, a contradiction. Hence we have q = p, and similarly, p = p; i.e., p = q = p>0. Consider the vector k ' E ~ defined by subtracting the two rows of N corresponding to sets S~ and T~ from the last row of N. We claim that the support of k contains that of k'. For coordinates corresponding to s, t and y variables this is clear. If coordinate j corresponds to a z variable and j is in the negative support of k', then j E St n Tq. Since Tt _D T2_D ... D Tq, we also have that j E St n T~ n 7"2n . . . n Tq; hence j is also in the negative support of k. On the other hand, if coordinate j corresponding to some z variable is in the positive support of k', then j ~ St n ~Pq (here 0 = E --. U). Thus j StnS2n...nspnTq, and so j is also in the positive support of k. This establishes the claim, and since k is elementary, k must be a positive scalar multiple of k'. Hence after deleting Ssi, 1 <- i <-p, and t~, 1 < j < q, constraint (4) is of the form z ( E ) - z ( S O - z ( T q , SO >- ro(E) - rl(Sl)

--

(5)

rz(Tq).

If p = 0 (q = 0), the same analysis determines a constraint of the form (5) in which St = ~ (Tq = 0). Thus the polyhedron ~ ' is given by z ( E ) - z ( S ) - z ( T , S ) >- ro( E ) - r,( S ) - r2( T ) ,

S, T C _ E ,

zs>-O,

j E E.

(6)

Since r2( T ) >- r2( T -- S ) , an equivalent representation is z ( E ) - z ( S ) - z ( T ) >- ro(E) - r t ( S ) - r2(T),

S, T C _ E , S N T = O ,

zs>-O,

j E E.

(7)

The latter system may be rewritten as z ( E ) - z ( U ) >- ro(E) - min {q(U') + r2( U - U')}, U'CU

u_c~,

zj_>0,

j~.

(8)

204

H.-C. Huang and L.E. Trotter Jr./Blocking and anti-blocking polyhedra

Edmonds has shown (see [3]) that for any set U C_E, ro(U) = rain {rs(U') + r , ( U ~ U')}. U'~U

Thus the following system determines the polyhedron ~ ' , z ( E ) - z ( U ) > ro(E) - to(U),

U C_ E,

zi -> 0,

j ~EE.

(9)

An analysis similar to that above, using part (ii) of Theorem 2 rather than part (i) of that theorem, yields the following anti-blocking polyhedron ti' for the polyhedron ~t' = {z E R~f: M ' z <- 1}, z(U) -< r0(U),

ucE,

z ( S tq T) < rl(S) + r2(T) - ro(E),

S, T C _ E and S U T = E,

zi>O,

]~E.

(10)

Of course, in this case we insist that M have no zero columns; i.e., so that no z-column of M is zero we require that each element of E be in a common independent set of size r0(E), and so that no s or t-column of M is zero, no element of E can be in all common independent sets of size ro(E). The inequalities (10) defining M' were first determined by McDiarmid [13]. One may obtain blocking and anti-blocking results for common independent sets of fixed cardinality r <- ro(E) simply by truncating (see [3]) the matroids at value r, i.e., by using rank functions r'i(S) = min(ri(S), r), S C_ E, i = l, 2. It is perhaps worth reemphasizing the generality and the limitations of the technique used here. In the present instance the success in determining relations (9) and (10) derived mainly from two factors: (i) knowing at the outset a polyhedral representation (1) for the combinatorial family of interest and (ii) applying submodularity of the matroid rank functions in order to determine nesting of the sets Si and Tj, as described following relation (4'). Item (i) above allowed us to invoke Theorem 2, whereas item (ii) led to the eventual simplification of (4) into the more meaningful form (9). Perhaps a similar analysis for other combinatorial families will prove useful in determining additional blocking and anti-blocking polyhedra.

References

[1] R.G. Bland, "Elementary vectors and two polyhedral relaxations", Mathematical Programming Study 8 (1978) 159-166. [2] W. Cunningham,"An unbounded matroid intersection polyhedron", Journal o[ Linear Algebra and Its Applications 16 0977) 209-215.

H.-C. Huang and L.E. Trotter Jr. / Blocking and anti-blocking polyhedra

205

[3] J. Edmonds, "Submodular functions, matroids, and certain polyhedra", in: R. Guy, H. Hanai, N. Sauer and J. Schonheim, eds., Combinatorial structures and their applications, Proceedings of the 1969 Calgary International Conference (Gordon and Breach, New York, 1970) pp. 69--87. [4] J. Edmonds and F.R. Giles, "A rain-max relation for submodular functions on graphs", Annals of Discrete Mathematics 1 (1977) 185-204. [5] D.R. Fuikerson, "Networks, frames, blocking systems", in: G.B. Dantzig and A.F. Veinott Jr., eds., Mathematics of the Decision Sciences, Lectures in Applied Mathematics, Vol. 11 (Am. Math. Soc., Providence, RI, 1968) pp. 303-'335. [6] D.R. Fulkerson, "Blocking polyhedra", in: B. Harris, ed., Graph Theory and Its Application (Academic Press, New York, 1970) pp. 93-112. [7] D.R. Fuikerson, "Anti-blocking polyhedra", Journal of Combinatorial Theory 12 (1972) 50-71. [8] D.R. Fulkerson, "Blocking and anti-blocking pairs of polyhedra", Mathematical Programming 1 (1971) 168-194. [9] D.R. Fulkerson and D.B. Weinberger, "Blocking pairs of polyhedra arising from network flows", Journal of Combinatorial Theory 18 (1975) 265-283. [10] F.R. Giles, "Submodular functions, graphs and integer polyhedra", Thesis, Department of Combinatorics and Optimization, University of Waterloo (Waterloo, Ont., 1975). [11] H.-C. Huang, "Investigations on combinatorial optimization", Thesis, School of Organization and Management, Yale University (Cornell University, School of Operations Research and Industrial Engineering, Tech. Rept. No. 308 (Ithaca, NY, 1976)). [12] E. Lawler, "Matroid intersection algorithms", Mathematical Programming 9 (1975) 31-56. [13] C.J.H. McDiarmid, "Blocking, anti-blocking, and pairs of matroids and polymatroids", Journal of Combinatorial Theory (B) 25 (1978) 313-325. [14] L.E. Trotter Jr. and D.B. Weinberger, "Symmetric blocking and anti-blocking relations for generalized circulations", Mathematical Programming Study 8 (1978) 141-158. [15] D.B. Weinberger, "Investigations in the theory of blocking pairs of polyhedra", Thesis, Corneli University, School of Operations Research and Industrial Engineering, Tech. Rept. No. 190 (Ithaca, NY, 1973). [16] D.B. Weinberger, "Network flows, minimum coverings, and the four-color-conjecture", Operations Research 24 (1976) 272-290. [17] D.B. Weinberger, "Transversal matroid intersections and related packings", Mathematical Programming 11 (1976) 164-176.

Mathematical Programming Study 12 (1980) 206-213. North-Holland Publishing Company

CERTAIN KINDS OF POLAR SETS AND THEIR RELATION TO MATHEMATICAL PROGRAMMING* Jergen T I N D Aarhus Universitet, Aarhus, Denmark

Received 28 April 1977 Revised manuscript received 14 August 1979 In this paper is discussed some special polar correspondences, which are related to the Minkowski polarity, known from convex analysis. They represent a natural generalization of the concepts of blocking and antiblocking polyhedra, developed by D.R. Fulkerson and used in the study of certain problems in mathematical programming and combinatorics. Here emphasis will be placed on the study of necessary and sufficient conditions to ensure the validity of the polar correspondences in question. At the same time an economic interpretation will be given within the framework of activity analysis. Key words: Blocking Sets, Antiblocking Sets, Polarity, Activity Analysis.

1. Introduction

This paper describes a generalization of and a connection b e t w e e n the Minkowski polarity and the idea of blocking and anti-blocking polyhedra, developed by F u l k e r s o n in [2], [3] and [4]. This is done by the introduction of generalized ideas as blocking and antiblocking sets and functions. Together with them an economic interpretation is given along the lines of nonlinear activity analysis (see e.g. Williams [11] or K n u d s e n [6]). S o m e of the material has been presented earlier in Tind [8] and [9], but in the present d e v e l o p m e n t with some modifications. Finally a relation is included between some special blocking and antiblocking pairs.

2. Polar sets

We will first state the m o s t basic properties of the classic Minkowski polarity. Consider a closed, c o n v e x set C C_R", 0 E C. Let C* denote the polar set of C, i.e. C* = { x * E R" I x " x*-< 1,Vx E C}. * Paper presented at the IXth International Symposium on Mathematical Programming, Budapest, 1976. 206

J. Tind/Certain kinds of polar sets

207

It is seen that C* is also closed, convex and 0 ~ C*. Additionally, we have the so-called involutory property, i.e. C** = C,

(2.1)

which says that C is the polar of C*. For arbitrary sets C E R" we have the more general formula: C** = cl(conv(C U {0})),

(2.2)

where "cl" denotes the "closure" and " c o n v " denotes the "convex hull" (see Rockafellar [7, p. 125]).

3. Antiblocking sets

The idea is here to modify the definition of polar sets by intersecting the polar sets by a given set D. As the polar set is basically a set which contains all normals for the supporting hyperplanes of the original set, we may lose some normals by this intersection. In general, this destroys the involutory property. Therefore, we will impose some necessary and sufficient conditions in order to save the involutory property. Let B _CR n and D _CR n be nonempty sets. Define the antiblocking set/~ _CR ~ of B with respect to D as follows:

B-B*ND. The notion "antiblocking set" is used here, because it represents a generalization of antiblocking polyhedra, introduced and studied by Fulkerson (see e.g. [3]). The basic question is now to examine the conditions under which /~ = B,

(3.1)

i.e. when B is the antiblocking set o f / ~ with respect to D. In this case B a n d / ~ constitute a pair of antiblocking sets. It is remarked that if R ~ = D, we are back in the Minkowski polarity. Hence, we are actually dealing with a generalization of this polarity. The following two theorems give necessary and sufficient conditions for Eq. (3.1) to be valid.

Theorem 3.1. Let D be closed, convex and containing the origin. Then

~=B if and only if

208

J. Tind/ Certain kinds of polar sets B = cl(conv(B U D*)) n D.

Proof. We will first show that cl(conv(B** U D*)) = cl(conv(B U D*)).

(3.2)

Since B** _~ B, the inclusion "_D" is obvious. To show that cl(conv(B** U D*)) _Ccl(conv(B U D*)),

(3.3)

it is remarked that cl(conv(B U D*)) _Dcl(conv(B U {0})) = B**, as {0} _CD*. Additionally, cl(conv(B U D*))_~D*. Since cl(conv(B U D*)) is closed and convex, we obtain (3.3), and (3.2) is proved. For closed, c o n v e x sets P, Q c_ R n containing the origin we have (P O Q)* = cl(conv(P* u Q*)). See e.g. Rockafellar [7, Corollary 16.5.2]. H e n c e =

n

o

= (n*

n/9)* n

o

= cl(conv(B** U D*)) n D = cl(conv(B U D*)) n D, where the last equation follows by (3.2). Theorem 3.2.* L e t D be closed, convex and containing the origin. Then

if and only if there exists a closed, convex set C C_R ~ such that B = C n D and such that D* C_ C.

Proof. Let us first assume t h a t / ~ = B, and let C = cl(conv(B U D*)). Obviously, C is closed and convex, and C _~ D*. T h e o r e m 3.1 implies that B = C n D. This shows one direction of the theorem. Now assume that we have a set C such that B = C A D and D'C_ C. [It is remarked that in general C might here be different from the previous set cl(conv(B U D*)).] F r o m Theorem 3.1 it is now sufficient to show that B = cl(conv(B U D*)) n D.

(3.4)

Since B = B n D _Ccl(conv(B U D * ) n D, it is enough to show the reverse inclusion: * It is remarked that for polyhedra Theorem 3.2 is a special case of joint work by Julian Arfioz, Jack Edmonds and Victor Griffin.Personal communication with Julian Arfioz.

J. Tind/ Certain kinds of polar sets

209

B _~ cl(conv(B O D*)) O D. By assumption C _~ D*. Moreover, C _~ B. Since C is closed and convex, we obtain that C _Dcl(conv(B O D*)). Hence, cl(conv(B U D*)) n D c_ C N D = B, where the last equation follows by assumption. This implies (3.4), and the theorem is proved. The assumption D'C_ C in the theorem is equivalent to D D C*. This expresses in particular that all supporting hyperplanes for C have their normals contained in D. The defining linear forms are normalized (equal to 1).

4. Antiblocking functions Antiblocking functions present a development similar to the one just outlined for antiblocking sets. The starting point is here the concept of conjugate functions and especially the similar notion of polar functions [7]. Again, the basic concepts take place within a given set D, and here we are intersecting the subgraphs of concave functions by D. Consider a concave, non-negative closed function f(x):R~-->R+, where R~denotes the non-negative orthant, i.e. R~ = {x E R n I x - 0}. Let sub§ f E R n denote the non-negative subgraph of f(x), i.e. sub+ f = {(x, y) E R "+1 [ x -> 0 and 0 --- y -< f(x)}. Since f(x) is a non-negative function, it is uniquely determined by sub§ f. With the given specifications on f(x) it follows that sub§ f is a closed, convex set containing 0. Define the following function f(x*) : R~--> R+, f(x*) = sup{y* E

R I- x .

x* + y*f(x) --< 1, Vx -> 0}.

Call f(x*) the antiblocking function of f. f(x*) becomes non-negative and non-decreasing. Its subgraph is given by sub+ [ = {(x*, y*) E R ~+z I (x*, y*). (-x, y) --< 1 for all (x, y) E sub§ f} n {(x*, y*) E R n+l I (x*, y*) _>0}. This shows that f is also concave and closed. Let T denote the linear transformation T : (x, y)-->(-x, y). If D = R~+1, it is seen that sub+ f is obtained by the antiblocking relation with respect to D as follows: sub+ [ = T(sub+ f). We now additionally assume that f(x) is non-decreasing in each component.

J. Tind/ Certain kinds of polar sets

210

This implies that all supporting hyperplanes for T(sub+ f ) have their normals in D = D**. Hence, it is obtained by the same reasoning as in the proof of Theorem 2.1 that T(sub+ f ) = sub+ f. This shows that

7=f,

4.1)

i.e. f is the antiblocking function of

5. An economic interpretation For a pair of conjugate convex functions Williams in [11] gives an economic interpretation of the corresponding involutory property. Along similar lines an economic interpretation is here to be given for a pair of antiblocking functions and the Eq. (4.1). It is remarked that [ ( x * ) can be expressed alternatively as f(x*) = inf x* 9 x + 1 ~-_o

f(x)

'

where (x* . x + 1)/f(x) = ~, if f ( x ) = O. The polarity will not be disturbed by rescaling, i.e. by replacement of the number 1 by an arbitrary number k > 0, which means that f ( x * ) = inf x 9 x* + k x~0

f(x)

Assume now that a manufacturer produces a product by means of n activities. Let the components of the vector x -> 0 denote the activity level of each activity. With a given activity level he produces f ( x ) units of the product. Assume additionally that the components of the vector x* -> 0 denote market prices that equal the cost for use or consumption of one unit of the corresponding activities. H e n c e x 9 x* is a cost for production of f ( x ) units of the product. In addition to this cost, which is linear in x, there is supposed to be a constant cost of size k. It is further assumed that the manufacturer's objective is to minimize the average cost per unit produced, i.e. the manufacturer wants to solve the following problem f ( x * ) = inf x 9 x* + k x~-o

f(x)

Hence, the antiblocking function of f denotes the minimal average cost, given a price vector x*. Now the manufacturer also considers selling his activities at a given level

Z Tind/Certain kinds o/polar sets

211

x-> 0 to a contractor, who in return should pay him with an amount of the finished product. For that purpose the contractor quotes a unit price x*-> 0 on each activity. Based on this price the manufacturer at least would demand an amount of the finished product that equals the estimated production costs, divided by the average cost per unit, i.e.

x .x*+k

[(x*) Hence, the contractor, seeing no reason to return more than that amount, will get the task to find a price x* -> 0 that solves the problem f ( x ) = inf x - x* + k x.~o [(x*) By (4.1) we get the reasonable result that with such a price the amount of finished product does not depend on whether the manufacturer produces by himself or lets the contractor do it for him. The main difference between the development here and that in [11] is that the manufacturer in [11] minimizes his total costs, whereas he here is minimizing the average costs. Additionally, the whole thing here is considered over the nonnegative orthant D = R~. This has the effect that x and x* are non-negative and that [(x) and [(x*) are non-decreasing functions, which seems reasonable in the economic context above.

6. Reverse polar sets The difference between the material on polar sets and the following is due to a change in the direction of the inequality sign in the basic definition. Consider a non-empty set C C R n. Let C O denote the reverse polar set of C, i.e.

C ~176

Ix" x~ > l, Vx E C}.

It is seen that C o is convex, closed and 0 ~ C. Additionally, C o is non-empty if and only if cl cony C ~ O. Let Po(C) denote the cone generated by C with vertex at O, i.e.

Po(C) = {Ax I x E C and A -> 0}. It can then be shown that C oo = cl conv C + P0(cl conv C). See [8, T h e o r e m 5.1]. This equation is corresponding to Eq. (2.2) for polar sets. If C = cl conv C + P0(cl conv C) and 0 ~ C, then the c o n v e x set C is called reverse closed, or by the terminology by Ar~oz for polyhedra/3-closed [1].

212

J. Tind/ Certain kinds of polar sets

For C reverse closed we have that

C00= C, which corresponds to the similar Eq. (2.1) for polar sets.

7. Blocking sets Here we will make a development similar to the one for antiblocking sets. Let B _CR n and D _CR" be nonempty sets. Define the blocking set/~ _CR" of B with respect to D as follows:

~-B~ Again, the notion "blocking" is due to Fulkerson [2]. We could now in full detail proceed along the same lines as in Section 3, but instead we refer to [8], where it has been done for certain sets. There are many similarities between antiblocking and blocking sets. One of them may be illustrated by the following observation. Let D = {x, y) ~ R ~-~ x R [ (x, y) > 0} be the positive orthant in R ~. Let B ~ R ~ a n d / ~ E R ~ be a pair of antiblocking sets with respect to this D. Let (XA,YA) E B, where (XA,YA) E R n-I • R, and let (x*, y*) E/~, where (X,~, y.~) E R n-I X R. Define the relation (XA, YA)--) (XB, YB) where xa YA and 1 YB = - -

YA

for

(XA,YA) E B,

and the relation

y*) (x~ yO) where

xo=XA y* and

1 Y~ = y---~A for (x'A, y~) ~/~. Then the two sets of points {(xB, YB)} and {(x~, y~)} constitute a pair of blocking sets with respect to D. This is due to the fact that the following two basic inequalities are equivalent for the present choice of D:

J. Tind/ Certain kinds of polar sets

213

XA " X~ + YAY~ <- 1

and x8 9 x ~ + ysyO _> 1. The differences between blocking and antiblocking sets b e c o m e clear, for example in the literature of the polyhedral case. See the next section.

8. Polyhedra IfD=R~={xER n l x - > 0 } a n d C = { x E R n l A x < - l } , w h e r e A i s a n m by n matrix of non-negative elements, and 1 ={1 . . . . . 1) with m elements, then T h e o r e m 3.2 can be applied on B = C n D. In this case B a n d / ~ constitute a pair of antiblocking polyhedra. There is a similar connexion to blocking polyhedra. Both c o n c e p t s play an important role in m a n y extremal combinatorial problems. See the original papers by Fulkerson [2, 3, 4] and the later papers, Fulkerson and Weinberger [5] and Weinberger [10], a m o n g others.

References [1] J.A. Ar,qoz, "Polyhedral neopolarities", Dissertation, Department of Applied Analysis and Computer Science, University of Waterloo (Waterloo, Ont., 1973). [2] D.R. Fulkerson, "Blocking polyhedra", in: B. Harris, ed., Graph theory and its applications (Academic Press, New York, 1970) pp. 93--112. [3] D.R. Fulkerson, "Anti-blocking polyhedra", Journal o[ Combinatorial Theory 12 (1972) 50-71. [4] D.R. Fulkerson, "Blocking and anti-blocking pairs of polyhedra", Mathematical Programming 1 (1971) 168-194. [5] D.R. Fulkerson and D.B. Weinberger, "Blocking pairs of polyhedra arising from network flows", Journal of Combinatorial Theory 18(B) (1975) 265-283. [6] N.C. Knudsen, Production and cost models o[ a multi-product firm (Odense University Press, Odense, Denmark, 1973). [7] R.T. Rockafellar, Convex analysis (Princeton University Press, Princeton, NJ, 1970). [8] J. Tind, "Blocking and antiblocking sets", Mathematical Programming 6 (1974) 157-166. [9] J. Tind, "On antiblocking sets and polyhedra", in: Hammer et al., eds., Studies in integer Programming, Annals of Discrete Mathematics 1 (1977) 507-516. [10] D.B. Weinberger, "Network flows, minimum coverings, and the four-color conjecture", Operations Research 24 (1976) 272-290. [1 I] A.C. Williams, "Nonlinear activity analysis", Management Science 17 (1970) 127-139.

Mathematical Programming Study 12 (1980)214-221. North-Holland Publishing Company

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