Submodular Functions and Optimization

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T h e o r e m 3.26 (The extreme ray theorem) [Tomi83]: The extreme rays of the characteristic cone C(/) of the base polyhedron B(/) are exactly those represented by the vectors Xe ~ Xe' (3.101) for all e, e' € E such that e covers e' in V = (E, <). (Proof) Since the characteristic cone C(/) is the set of the boundaries of nonnegative flows in M = (G = (E, A), c) defined above, C(/) is generated by vectors Xe — Xe' such that (e, e') £ A. Moreover, vector Xe — Xe' fc>r an arc (e, e') € A can not be expressed as a nonnegative linear combination of the other vectors % ei — Xe' with (ei,e[) £ i - {(e, e')}. Q.E.D. Theorem 3.13 characterizes the dual cone C*(/) of C(/), where C*(/) = {y | y € HE, Vx € C(/): £

x(e)y(e) < 0}.

(3.102)

e£E

Theorem 3.13 says that — C*(/) consists of monotone nondecreasing functions from V = (E, -<) to (R, <) or that C*(/) consists of monotone nonincreasing functions from V = (E,^~) to (R, <). Therefore, C*(/) is generated by the characteristic vectors of the (lower) ideals of V and — XE(b) Elementary transformations of bases For any base x of submodular system (T>, f) on E and for e, e' £ E such that e' £ dep(x, e) — {e} we have x + a(Xe-Xe')£B(f)

(0
(3.103)

The transformation of base x € B(/) into such a base x + a(xe — Xe') ^ B(/) is called an elementary transformation of base x £ B(/). The following theorem is important from an algorithmic point of view. For any real number a, [a\ denotes the maximum integer less than or equal to a. Theorem 3.27: For any two bases x, y £ B(/) base x can be transformed into base y by at most [|_E|2/4J repeated elementary transformations such that each component x(e) with x(e) < y(e) monotonically increases and each component x(e) with x(e) > y(e) monotonically decreases. (Proof) Consider the following algorithm.

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71

1° If x = y, then stop. 2° Choose any element e G E such that x{e) < y(e). 3° Choose any element e! G dep(x, e) such that x(e') > y(e'). Put a <— min{y(e) - x(e), x(e') - y(e'), c(x, e, e')} and x <- x + a ( x e - Xe') 4° If a < y(e) - x(e), then go to Step 3°. Otherwise (a = y(e) — x(e)) go to Step 1°. Note that if x ^ y, there is an element e such that x(e) < y{e) and that if x(e) < y(e), there is an element e' G dep(x, e) such that x(e') > y(e'), since otherwise, putting X = dep(x,e), we would have f(X) = x(X) < y(X), a contradiction. Also note that if a < y(e) — x(e) in Step 4°, we have a = min{x>(e/) — y(e'), c(x, e, e')} and the number of elements in {e" | e" G dep(x,e), x(e") > y(e")} decreases by at least one after the elementary transformation x <— x + a(Xe — Xe')- Therefore, the case where a < y{e) — x(e) is repeated at most \S~\ times, where S~ = {e \ e € E, x(e) > y(e)} for the initial base x. Denning S+ = {e \ e G E, x(e) < y(e)} for the initial x, the total number of the elementary transformations is at most \S+\ x | S - | , which is bounded by LI-^lV4]Q.E.D. Consider a capacitated network J\f = (G = (V, A),c,c) with an underlying graph G and lower and upper capacity functions c, c: A —> R with c < c. Let KCJC- 2V —> R be the cut function associated with the network J\f = {G = (V, A),c,c) (see (2.62) in Section 2.3). The set of the boundaries of feasible flows in J\f is the base polyhedron associated with the submodular system (2V, nc,c)- Note that there exists a feasible circulation (a feasible flow ip such that dip = 0) in J\f if and only if 0 G B(«c,c)- Since dc G B(KC,C), there exists a feasible circulation in M if and only if the base dc € B(KCJC) is transformed into 0 by repeated elementary transformations as in Theorem 3.27. A standard algorithm for finding a feasible circulation by the use of a max-now algorithm consists of such repeated elementary transformations (cf. [Hoffman60], [Ford+Fulkerson62]). For a (directed) graph G = (V, A) and a {0,1}-valued function ip: A —> {0,1} define the graph G^ as the one obtained from G by reorienting arcs a G A such that ip(a) = 1. We say p defines the reorientation G^. A graph G = (V, A) is strongly k-connected (k: a positive integer) if for each nonempty proper subset U of vertex set V there exist at least k arcs from

72

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U to V — U. We call {0,1} a strongly k-connected reorientation of G if G

R by c(a) = 1

(a £ A),

(3.104)

and let K: 2 y —> R be the associated cut function, i.e., K(U) = c(A+U) = \A+U\ for U C y . Also define (fe)

K

((7j

f K(C/)-fc

" j o

(C/€2^-{0,y})

(Ue{0,V}).

[6AUb)

Then K> >: 2V —> R is a crossing-submodular function on 2^ and defines a base polyhedron B(K^), if B(K^) ^ 0, due to Theorem 2.5. B ( K ^ ) is called the k-abridgment of B(K) in [TomiSlb, 81c]. It can easily be seen that B(re' ^) = {9(y3 | 93 defines a strongly ^-connected reorientation of G}. (3.106) Here, we assume that the underlying totally ordered additive group R is the set Z of integers. Also note that the strong ^-connectedness of a reorientation of G depends only on the boundary dip of

Note that an elementary transformation of a base (or a boundary dip) in B(«;(fc)) corresponds to a transformation of the reorientation of G (defined by ip) by reversing the arcs in a directed path and possibly directed cycles and that reversing arcs in a directed cycle does not change the boundary dip.

(c) Tangent cones For any base x € B(/) associated with submodular system (P, / ) on E the tangent cone of B(/) at x, denoted by TC(B(/),x - ), is defined by TC(B(/),x) = {Ay | A > 0, y e RE, x + yG B ( / ) } .

(3.107)

3.3. Structures of Base Polyhedra

73

Here, the underlying totally ordered additive group R is assumed to be the set of reals (or rationals). Given a base x G B ( / ) , we call an ordered pair (e, e') of elements of E an exchangeable pair associated with x if e! G dep(x, e) — {e}. Theorem 3.28: The tangent cone TC(B(/),z) of B(/) at a base x is generated by the set of the following vectors: Xe

~ Xe> (e € E, e' G dep(x, e) - {e}).

(3.108)

In other words, for any vector y G TC(B(/), x) there exist some nonnegative coefficients A(e, e') for exchangeable pairs (e, e') such that y = ^J{A(e, e'){Xe~Xe') I (e>e ' ) :

an

exchangeable pair associated with x}. (3.109)

(Proof) Let C be the cone generated by the vectors in (3.108). It follows from the definition of dependence function that C C TC(B(/),s).

(3.110)

Suppose that there exists a vector y G TC(B(/),x) — C. Then, by the separation theorem (Theorem 1.13) there exists a vector w G R such that VzGC: J2 w{e)z{e) > 0,

(3.111)

eeE

X>(e)y(e)<0.

(3.112)

e€E

From Theorem 3.16 and (3.111) the base x is a minimum-weight base with respect to the weight function w but (3.112) implies that for a sufficiently small a > 0 x+ay is a base and the weight of the base x+ay is smaller than that of x. This is a contradiction, so that we must have C = TC(B(/),x). Q.E.D. A constructive proof of this theorem for polymatroids was given in [Fuji78a, Lemma 9]. It should also be noted that Theorems 3.26 and 3.28 are closely related. We can show Theorem 3.28 by using Theorem 3.26 and vice versa. Suppose that (T>, / ) is a simple submodular system on E and that x is an extreme point of B ( / ) . Then, V(x) = {X \ X eV,

x(X) = f(X)}

(3.113)

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is also a simple distributive lattice since the length of a maximal chain of T>(x) is equal to \E\ due to Theorem 3.22. This fact is crucial in the following argument. Let V(x) = (E, <x) be the poset representing V(x). An efficient method for finding dep(x,e) for all e € E is shown in [Bixby+Cunningham+Topkis85] (for polymatroids). Their algorithm also discerns whether a given vector is an extreme base. The algorithm adapted for submodular systems is given as follows. Let x be any vector in ~RE. Algorithm Step 1: Put S1 ^ 0. Step 2: For each % = 1, 2,

, \E\ do the following (2-1) and (2-2).

(2-1) If there exists no element e € E — S such that S U {e} € V and x(S U {e}) = f(S U {e}), then stop (x is not an extreme base). Otherwise find one such element e € E — S and put S <— S U {e} and ei <— e. (2-2) Put T <- S. For each j = 1, 2, , i — 1 do the following (*). (*) If x(T - {a-j}) = f(T - {ei-j}), then put T ^ T - {e^}. Put dep(x, ei) <- T. (End) If x is an extreme base, then V(x) = 2V^ with V{x) = (E, ^x). Repeating Step (2-1), we have a maximal chain So C Si C C Sn (n = \E\) of T>(x), where Si = {ei, &2, , B{\. Note that a maximal chain of T>{x) is obtained by augmenting elements of an ideal S of V(x) one by one in Step (2-1). Conversely, if we find a chain 0 = So C Si C C Sn = E such that x(Si) = f(Si) (i = 1, 2, , n), then x is an extreme base due to Theorem 3.22. This validates Step (2-1). We now show the validity of Step (2-2). Because of the above argument we can assume that the given x is an extreme base. For the current i at the beginning of an execution of Step (2-2) we have a chain 5o C 5i C C Sj. Note that dep(x,ei)CSi, dep(x,ei)USi_i e D(z)

(j = 0,1,

(3.114) ,i).

(3.115)

3.3. Structures of Base Polyhedra

75

Since distinct members of (3.115) form a maximal chain from dep(x, ej) to S{, the final T obtained in the current Step (2-2) is dep(x, ej). The Hasse diagram G(x) = (E,A(x)) of V(x) = (E,^.x) can be constructed by the use of dep(z, ej) (i = 1, 2, , n). We can also modify the above algorithm so as to construct the Hasse diagram of V{x) = (E, ^ x ) while executing the algorithm. In Step 1 we put G(x) = (E,$). Modify Step (2-2) as follows. (2-2') Put T <- 5 and W <- 0. For each j = 1, 2, , i — 1 do the following (*1) and (*2). (*1) If d-j i W and x(T - {e,_,}) = f(T - {e^-}), then put T ^ T — {ej_j}. (*2) If a-j i W and x(T - {e^}) < f(T - {e^}), then add to G(x) an arc from ej to ej_j and put W <- W U dep(x-, ej_j). Put dep(x, ej) <— T. It should be noted that ej is adjacent to ej_j in the Hasse diagram if and only if ei-j G dep(x-, ej) and e-i-j is a maximal element of dep(x, ej) — {ej} in V{x). Set W in Step (2-2') is introduced due to the fact that ej_j € dep(x, ej) implies dep(x, ei-j) C dep(x, ej). Set W can also be incorporated into Step (2-2), which may reduce the number of evaluations of / .

(d) Faces, dimensions and connected components [Fuji84d] Consider a simple submodular system (T>, f) on E. In the following, R is the set of reals (or rationals). For any f C D define

F(^) = {x | x £ HE, VX € T: x(X) = f(X), WX € V - T: x{X) < f(X)},

(3.116)

F°(^) = {x | x G RE, VX G T: x(X) = f(X), V l G D - f : x(X) < f(X)}.

(3.117)

Also, define D = {Vo | Vo is a sublattice of V with 0, E G £>0, F°(D 0 ) / 0}- (3.118)

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L e m m a 3.29: The collection D of sublattices of V defined by (3.118) is given by D = {D{x) | x £ B(/)}, (3.119) where for each x 6 B(/) T>{x) is defined by (3.113). (Proof) If £>0 G D, then for any x € F°(£>0) we must have Vo = V[x) by definition (3.117). Conversely, for any x € B(/) V{x) is a sublattice of V with 0, £ € P(x) and x E F°(V(x)). Hence we have V{x) € D. Q.E.D. It should be noted that for each 2?o € D F(2?o) isa face of B(/) and F°(Po) is the relative interior of the face F(£>o)We see from Lemma 3.29 that D is the collection of equality sets, each given by T>(x), for B(/) expressed by x(X)
(leP),

(3.120)

x(E) = f(E).

(3.121)

From Lemma 1.8 we can easily show Theorem 3.30: The collection F of all the nonempty faces of B(/) is given by {F(V0) | T>0 £ D}. Also, (i) IfV1,V2eB

and Vx / V2, then F(2?i) / F(X>2)-

(ii) For any £>i,£>2 € D, £>i C 2?2 if and only ifF(V2)

C F(2?i).

In other words, F in (3.116) determines an anti-isomorphism from D onto F, where D and F are considered as posets relative to set inclusion. F (or FU{0} when F does not have a unique minimal element) is the face lattice of B ( / ) . For two posets V% = [E%,^ii) (i = 1,2) we say V2 is a homomorphic image of V\ if there exists a mapping i\) from E\ onto Ei (i = 1,2) with 0, E € P j , we .have X>i C P 2 if and only ifH(V2) is a refinement ofH(Vi) and V(T>\) =

3.3. Structures of Base Polyhedra

77

(II(Pi), T2V1) is a homomorphic image ofViT>2) = (n(P 2 )> ^v2) under the natural mapping (i.e., T2 £ n(Z>2) is made to correspond to T\ if T2 C Ti). (Proof) Suppose V\ C P 2 . For each i = 1,2, distinct elements e and e' of E belong to different components of TL(T>i) if and only if there exists a n J g D j such that |{e, e'}nX| = 1. Therefore, since T>\ C X>2, n(2?2) is a refinement of n ( P i ) . Also, since we have I^^pgT^ if and only if T'2 Q X E T>2 implies T2 C X and since Z>i C Z>2, T2 C X € P i implies T2 C I if T2
Q.E.D.

T h e o r e m 3.32: For T>0 £ D we iave dimF(Po) = \E\ - \n(V0)\,

(3.122)

wiiere dimF(Po) is the dimension of the face F(Po). (Proof) The dimension of the face F(Po) is equal to that of the affine set M ( P 0 ) = {x\xGHE,yX

G P o : x(X) = f(X)}.

(3.123)

Since the rank of the coefficient matrix in the right-hand side of (3.123) is equal to |n(P 0 )|, we have (3.122). Q.E.D. It may be noted that the extreme point theorem (Theorem 3.22) and the extreme ray theorem (Theorem 3.26) easily follow from Theorems 3.30 and 3.32 and Lemma 3.31. Lemma 3.33: Suppose VQ £ D and let C o : 0 = So C Si C

C Sk = E

(3.124)

be a maximal chain ofT>Q. Then, F(Co) = F(Vo).

(3.125)

(Proof) Since U(C0) = n ( P 0 ) , we have x £ F(C0) if and only if x £ F(P 0 ). Q.E.D. Theorem 3.34: Suppose Po € D. Then a base x £ B ( / ) is an extreme point of the face F(Po) if and only if, for a maximal chain C: 0 = Sb C Si C

C Sn = E

(3.126)

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ofV which contains a maximal chain ofVo as a subchain, x is given by x(Si-Si-1)

= f(Si)-f(Si-1)

(i = l,2,---,n).

(3.127)

(Proof) The "if" part: Let Co be a maximal chain of T>Q contained in chain C of (3.126). Then, from (3.127) x € F(C,). Hence, from Lemma 3.33, x £ F(VQ). Since x is an extreme point of B(/) due to Theorem 3.22, x must be an extreme point of F(Po). The "only if" part: For any extreme point x of F(Z>o) we have T>Q C P(x) € D. Since x is also an extreme point of B(/), i.e., {x} is a zerodimensional face F(P(x)) of B ( / ) , a maximal chain of P(x) is a maximal chain of P due to Theorem 3.32. It follows that a maximal chain of P(x) which contains a maximal chain of Po as a subchain is a desired maximal chain of P . Q.E.D. Also, extreme rays of a face of B(/) are characterized by the following. T h e o r e m 3.35: Suppose P o e D and U(V0) = {Ti\i<El}. Then the set of all the extreme rays of the characteristic cone of the face F(Po) is given by {Xe ~Xe> | e = d+a, e' = d~a, a € A*(T>), 3i € I: {e,e'} C Tt}, (3.128) where A*{V) is the arc set of the Hasse diagram H(V(V)) representing the poset V(T>) = (E, -<x>)-

=

(E,A*(V))

(Proof) The characteristic cone C(Po) of the face F(Po) is given by C(2? o ) = {x\xeKE,

V I G Vo: x(X)

= 0,

V l f E D - Vo: x(X) < 0}.

(3.129)

It follows from (3.129) and the extreme ray theorem (Theorem 3.26) that the set of the extreme rays of C ( / ) ( = C({0, E})) which belong to C(DQ) is exactly given by (3.128). Q.E.D. We say that a submodular system (P, / ) on E is connected if there is no nonempty proper subset X of E such that X,E — X £ V and f(X) +

f(E-X) = f(E). T h e o r e m 3.36: A submodular system (P, / ) on E is connected if and only if {0, E} £ D, i.e., there exists a base x € B(/) such that x(X) < f(X)

(3.130)

3.3. Structures of Base Polyhedra

79

for any X € P with 0 ^ X ^ E. (Proof) The "if" part: If there exists a base x £ B(/) satisfying (3.130) for each X eV with 0 / X / £ , then /(£) = z(£) = x(X) + x(E -X)< f(X) + f(E - X)

(3.131)

for each X € P with E - X E V and 0 ^ X ^ E. The "only if" part: Suppose that (V, f) is connected. We have B(/) = F(Z>o) for some Vo £ D . We show Vo = {$,E}. Note that we have {0, E} C DQ. Suppose that there exists some Y € T>o such that 0 ^Y ^ E. Then, for any base x we must have Ve <E E - Y: dep(x, e)CE-Y

(3.132)

since otherwise there would exist a base x' such that x'("K) < f(Y), which would contradict the fact that Y € VQ. It follows from (3.132) that E-Y<EV,

x(E-Y) = f(E-Y).

(3.133)

Since x(Y) = f(Y), we have from (3.133) f(Y) + f(E -Y) = x{Y) + x{E -Y) = x{E) = f(E).

(3.134)

This implies that (P, / ) is not connected, which contradicts the assumption. Q.E.D. Therefore, we must have {0, E} = T>Q. The proof of the "only if" part shows the following. Lemma 3.37: Suppose that B(/) = F(P 0 ) for some P o € D. Then P o is a Boolean sublattice ofT>. Theorem 3.38: For a submodular system (P, / ) on E there uniquely exists a partition P* = {Ti, T2, , Tk} (k > 1) of E such that (i)T,GP

(i = 1,2,

,

(ii) each reduction (PT% / T j ) = (P, / ) T{ (i = 1,2, and

(ih) f(x) = / T '(T 1 nx) + - + fT"(Tk nx)

, k) is connected,

(xe v).

(3.135)

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(Proof) Suppose that Vo € D satisfies F(£>0) = B(/). From Lemma 3.37, T>0 is a Boolean lattice. Let k be all the minimal nonempty elements (as subsets of E) of T>o, i.e., Tl(T>o) = {Ti, , Tk}. Then, TiEV

(i = 1,2,

.

(3.136)

Since

kf{E)

= f(Tl) + f(E-T1) + --- + f(Tk) + > f(T1) + --- + f(Tk) + (k-l)f(E),

f(E-Tk) (3.137)

we have

f(E)

= f(T1) + --- + f ( T k ) .

(3.138)

For any X € V we have from (3.138)

f(X) + f(E)

= f(X)+f(T1) + --- + f(Tk) > /(Ti n X) + + f(Tk DX) + f(E) > f(X) + f(E).

(3.139)

It follows from (3.139) that

f(x) = f(T1nx)

+ --- + f(Tknx)

(xev).

(3.uo)

Also, for each Tj (i <E {1, 2, , k}) and X C Tj such that I T, - I G P , we have from (3.140) with X replaced by E - X

f(E)

< f(X) + f(E-X) = f(T1) + --- + f(Ti.1) + f(X) + +f(Ti+1) + --- + f(Tk),

6 P and

f(Ti-X) (3.141)

where (3.141) holds with equality if and only if f(Ti) = f(X) + f(Ti-X).

(3.142)

If (3.141) holds with equality or (3.142) holds, X must be a member of VQ and hence either X = 0 or X = Ti by the definition of T{. Therefore, (T>, / ) T{ is connected. It follows that the partition {Ti \ i = 1, 2, , k} derived from VQ satisfies (i)~(iii) of the present theorem. Moreover, if a partition {Wj \ j € J } satisfies (i)~(iii) with Tj's replaced by Wj's, then from (iii) we have Wj € VQ for each j € J. If {Wj | j €

3.3. Structures of Base Polyhedra J } ^ {Ti | i E / } , there exists Wj such that Wj = Th U ii, , ip G / with p > 2. Then, we have

81 U Tip for some

f ( W j ) = f ( T i l ) + --- + f ( T i p ) ,

(3.143)

from which follows that (T>, f) Wj is not connected. Consequently, we have {Wj | j G J } = {Tj | i € / } , which shows the uniqueness of the partition. Q.E.D. Theorem 3.39: Il(2?o) is the partition of E given by the decomposition of (V, f) into connected components. In particular, the number of connected components is equal to |II(2?o)|(Proof) The present theorem follows from the proof of Theorem 3.38. Q.E.D. From Theorems 3.32 and 3.39 we have Corollary 3.40: Let n* be the number of the connected components of (VJ). Then, dimB(/) = |£|-n*. (3.144)

The dimension of B(/) was also given in [Shapley71], [Giles75], [Bixby+ Cunningham+Topkis85] and [Tomi83] (when V = 2E). The following lemma is useful for finding the partition n(2?o) ( s e e [Bixby+Cunningham+Topkis85]). Lemma 3.41: For any base x G B(/) let G(x) = (E,A(x)) be the graph with vertex set E and arc set A(x) = {(e, e') \ e' G dep(x, e)}. Then, II(Po) is the set of the vertex sets of the connected components ofG(x). (Proof) Let {Wj \ j G J } be the set of the vertex sets of the connected components of G{x). Since Vo C V{x) = {X \ X G V, x(X) = f(X)}, partition {Wj \ j E J} is a refinement of U(T>Q). However, for any j' E J we have Wj, E — Wj G V(x) by the definition of G{x) and Wj, and f{E) = x{E) = x{Wj) + x(E - W3) = f{Wj) + f(E - Wj).

(3.145)

Hence, Wj G VQ for any j G J. Therefore, n(2?o) is a refinement of {Wj | j G J } . It follows that n(X>0) = {Wj \ j G J } . Q.E.D.

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Since, in particular, Lemma 3.41 holds for an extreme base x £ B(/), the connected components of (P, / ) can efficiently be found by the use of the algorithm shown in Section 3.3.C [Bixby + Cunningham + Topkis85]. Theorem 3.42: For any P i , P 2 £ D wehaveT>iC\V2 € D andF(PinP 2 ) is the unique minimal face which contains both F(Pi) and F(T>2). Moreover, if F(T>i U P 2 ) 7^ 0j then F(Pi U P 2 ) is the unique maximal face which is contained in both F(Pi) andF(V2). P 3 € D satisfying F(P 3 ) = F(PiUP 2 ) is the unique minimal sublattice ofV in D which contains both T>\ and P 2 . (Proof) The present theorem follows from the general theory of convex polyhedra (see Lemma 1.8). Q.E.D. It should be noted here that the unique minimal sublattice of P which contains both P i and T>2 does not necessarily belong to D. The following theorem gives a necessary and sufficient condition for a sublattice Po of P to be a member of D. Theorem 3.43: Let Po be a sublattice of P with 0, E £ Po and /o be the restriction of f to VQ. Then Po € D if and only if the following three statements hold: (i) /o is a modular function. (ii) For any maximal chain Co: 0 = So C 5i C of Po each minor (P, / ) Si/Si-i

C Sk = E

(i = 1,2,

(3.146)

, k) is connected.

(iii) Let Co be any maximal chain of Do as in (3.146) and x be any base of (Po,/o) such that x(Si) = MSi)

(i = 1,2,

)

(3.147)

(i.e., Co Q T>o(x))- Then for any X € P such that (a) X is compatible with n(P 0 ) and (b) x(X) = f(X), we have X £ P o . (Proof) The "if" part: Suppose (i)~(iii) hold. For each i = 1,2, , k let x* be a base of (Pj,/j) = (P, / ) Si/Si-i such that for any X £ P with Si-i C X C Si we have x*{X - 5i_i) < f{X) - /(5i_i) = /i(X - 5i_i).

(3.148)

3.3. Structures of Base Polyhedra

83

Such a base x* exists due to Theorem 3.36. Let x* € HE be the direct sum of x* (i = 1,2, , k), i.e., x*(e) = x*(e) (3.149) for each e € E with e € S{ — S{-i and i € {1, 2, , A;}. Then we have x* € B ( / ) due to Lemma 3.1. We show x* 0), which will complete the proof of the "if" part. Let X be any member of VQ. Since / is modular on VQ and x* satisfies x*(Si) = f(Si)

(i = 1,2,

)

(3.150)

on the maximal chain of VQ given by (3.146), we have

x*(X) = f(X)

(X € Vo),

(3.151)

where recall that a simple submodular system with a modular rank function has a unique extreme base. Now, suppose X € V — VQ. If X is compatible with H(VQ), then from (iii) we have x*{X) < f(X). (3.152) On the other hand, if X is not compatible with H(T>Q), i.e., for some io € {1, 2, , k} X - Sio + 0 and Sio+1 then denning T{ = St - 5i_i (i = 1, 2, , k), we have /(X)-x*(X)

= f(X)-x*(X) + Y^{f(Si)-x*(Si)} i=l ifc

> 5]{/((x n 5,) u 5i_i) - x*((x n 5,) u 5,_i)} = Ei/((^

nT

0 u 5 i-i) - / ( ^ - i ) - x*(x

> 0

nT

^)} (3.153)

due to (3.148) and (3.149). Therefore, VX € V - Vo: x*(X) < f(X). From (3.151) and (3.154) we have x* € F°(X>0)-

(3.154)

84

II. SUBMODULAR

SYSTEMS AND BASE

POLYHEDRA

The "only if" part: Suppose VQ € D. Then there exists a vector x' £ F°(X>o), from which follows (i). Also (ii) follows from Theorem 3.36. We show (iii). Let x be any base of (£>o>/o) such that (3.147) holds. Suppose that XQ € V is compatible with II(X>o) and that x(X0) = f(X0).

(3.155)

Since x' G F°(T>o) also satisfies x'(Si) = f(Si)

(* =

,

(3.156)

we have from (3.147) and (3.156) x{T) = x'(T)

(Ten(Po)).

(3.157)

Since XQ is compatible with U.(T>o), it follows from (3.155) and (3.157) that x'(Xo) = x{X0) = f(X0).

(3.158)

Since x' € F°(P 0 ), we must have Xo € Vo.

Q.E.D.

From Theorems 3.32 and 3.43 we have the following. Corollary 3.44: Suppose that B(/) is full-dimensional, i.e., dim B(/) = E\ - 1, and let Vo be a sublattice ofV with 0, E G Vo. Then, F(V0) is a facet of B(/) if and only if 2?0 forms a chain 0 = 5"0 C Si C S2 = E of V (of length 2) and (V,f) Si/Si-! (i = 1,2) are connected. We also have T h e o r e m 3.45: Let P i be a sublattice ofV with 0,E £V\. f is modular on V\. Also, let Cr. 0 = S o c S i C

k

=E

Suppose

that

(3.159)

be any maximal chain ofV\ and for each i = 1, 2, , k let Pi = {T^,T?, , 1 T[ } be the partition of Si — Si-\ given by the decomposition of (P, / ) Si/Si-! into connected components. Define P* = Ui=i Pi and let x be a base of (D, f) such that x{Sl)

= f{Sl)

(i = 1,2,

.

(3.160)

3.3. Structures of Base Polyhedra

85

Then T>2 given by

V2 = {X | X € P, x(X) = f(X), X is compatible with P*}

(3.161)

belongs to D and satisfies F(2?i) = F(Z>2).

(3.162)

(Proof) We can easily see that P 2 given by (3.161) satisfies (i)~(iii) of Theorem 3.43, where note that the set of minors (P, / ) Si/Si-i (i = 1, 2, , k) does not depend on the choice of a maximal chain since / is modular on Pi (see Theorem 7.17). Hence we have P 2 € D due to Theorem 3.43. Moreover, for any x' E F(Pi) we have x(Tl) = x'(T/)

(j = 1, 2,

, n; i = 1,2,

, k).

(3.163)

It follows that for any X E P 2 x'(X) = x(X) = f(X),

(3.164)

i.e., x' E F(P 2 ). We thus have F(PX) C F(P 2 ). On the other hand, since / is modular on Pi, from (3.160) and (3.161) we have Pi C P 2 and F(P 2 ) C F(Pi). Hence F(P 2 ) = F(X>i). Q.E.D. Note that the operation of getting P 2 from Pi in Theorem 3.45 is a closure operation on sublattices of P , containing 0 and E, on which / is modular and that D is the collection of closed sublattices of P . Theorem 3.45 is closely related to the concept of f-skeleton introduced in [Nakamura + Iri81]. An /-skeleton is a sublattice of P on which / is modular. A sublattice P i of P in Theorem 3.45 is an /-skeleton and there exists a unique maximal /-skeleton P* (maximal with respect to set inclusion) such that n(P*) = n ( P i ) , which is called the maximal /-skeleton corresponding to P i ([Nakamura + Iri81]). We can easily see that P* may not be closed and that P* C P 2 , where P 2 is the closure of Pi given by (3.161). In fact, Pf is the aggregation of P 2 by U(T>i). Therefore, P 2 is a finer structure than the maximal /-skeleton P*. Theorem 3.45 gives a polyhedral interpretation of skeletons. Theorem 3.46: Let x be an extreme point of B(/) and K be an edge of B(/). Suppose {x} = F(P 0 ) and K = F(Pi) for P 0 , P i € D. Then, we have x E K (i.e., x is an end point of K) if and only if for {e, e'} £ n ( P i )

86

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

(i) vertices e and e' in the Hasse diagram R(V(V0)) ) are adjacent and

= (E,A*(V0)) of

(ii) the Hasse diagram H(T(T>i)) ofV(V\) is obtained by identifying e with e! in H('P(£>o)) (i.e., contracting the arc which connects e and e')

(For the definition of V{T)i) {% = 0,1) see Section 3.2.a.) (Proof) From Theorem 3.32, we have | n ( P 0 ) | = \E\ and | n ( P i ) | = \E\ - 1, so that LI(Pi) consists of a two-element set and singletons. From Theorem 3.43 and Lemma 3.31, we have x £ K if and only if (1) T>\ C P o and (2) for T € n ( P i ) with | r | = 2 and for Su S2 € P i with Si C S2 and T = S2 - Si (V, / ) S2/S1 is connected. The present theorem follows from this fact. Q.E.D. Theorem 3.47: Let Xj (i = 1,2) be distinct extreme points of B ( / ) and suppose {xi} = F(T>i) for D, G D (i = 1,2). Then xi and x2 are adjacent in B ( / ) if and only if there exist vertices e and e' in E connected by an arc both in H(V(T>i)) and in H(T(V2)) such that the Hasse diagrams obtained by identifying e with e! in H(V(T>i)) and H(V{T>2)), respectively, coincide with each other. (Proof) From Theorems 3.32 and 3.43 and Lemma 3.31 we see that extreme bases xi and x2 are adjacent in B ( / ) if and only if the following two statements hold: (i) There exists a common sublattice P3 of T>i and Z>2 such that 0 , B G V3 and | n ( P 3 ) | = | ^ | - 1 . (ii) For T <E n ( P 3 ) with \T\ = 2 and for SUS2 € P3 with Si C S2 and T = S2 — Si (P, / ) S2/S1 is connected, from which follows the present theorem.

Q.E.D.

It should be noted that for extreme bases X{ (i = 1, 2) we have P(XJ) £ D (i.e., T>{ = P(xj) in Theorem 3.47) and that if xi and x2 are adjacent, we have P ( x i ) n P ( x 2 ) € D and F(P(xi) n P ( x 2 ) ) is the edge of B ( / ) incident to xi and x2. 3.4. Intersecting- and Crossing-Submodular Functions

3.4. Intersecting- and Crossing-Submodular Functions

87

In this section we prove Theorems 2.5 and 2.6 presented in Section 2.3 which relate intersecting- and crossing-submodular functions to submodular systems ([Fuji84b]). (a) Tree representations of cross-free families We first introduce the tree representation of cross-free families due to Edmonds and Giles [Edmonds+Giles77]. We call a set {Xi | i € / } of subsets X{ (i 6 /) of E a copartition of E if {E — Xi \ i € / } is a partition of E. Let T = (V, A) be a tree with a vertex set V and an arc set A = {ai \ i £ / } . Also let V = (Pv | v 6 V) be a family of subsets of E such that the nonempty P v 's form a partition of E. Possible repeated members in V are the empty set only. Note that each vertex v E V is given a subset Pv of E. Deleting an arc a,{ (i G /) from tree T yields a graph with two connected components. Denote the vertex set of the connected component which contains the tail d+a,i of a\ by V(aj), and define

% = [}{pv I v G V(ai)}.

x

(3.165)

We thus have a family F=(Xi\i£l)

(3.166)

of subsets of E. From this construction we can easily see that the family J- is cross-free. We call the representation of JF by the pair of the tree T = (V, A) and the family V = {Pv \ v G V) a tree representation of T. We show the converse that every cross-free family has one such tree representation. First, consider a laminar family T = (X-i \ i € /) of subsets of E, i.e., for each Xi, Xj (i,j € /) we have at least one of the following three: (1) Xi DXj = 0, (2) Xi C Xj and (3) X,h D Xj. For simplicity suppose that Xi (i € /) are distinct. Let IQ be a new index not in / , and define a vertex set V by V = {Vi | i € / U { i 0 } } (3.167) and an arc set A = {ai | i e / } .

(3.168)

The incidence relation is defined as follows. For each i € / such that Xi is a nonempty and non-maximal member of T, let X^ be the unique minimal member of T which properly includes Xi. (Note that the members of T

88

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

which includes JQ form a chain due to the fact that T is a laminar family (without repeated members).) Then we define d+a>i = v% and d~ai = v^. For each maximal member X\ of T we define d+ai = v\ and d~cii = Uj0. Also, if JF contains Xj = 0, then for some nonempty minimal member X\ of T we define <9+aj = v\ and 9^aj = uj. Consequently, the graph T = (V,A) is a directed tree toward the root ViQ. We define Xi0 = E and for each i €/U{i0} p vi =Xi~ [ji^k \k£l, d-ak = Vi}. (3.169) We can see that the pair of tree T = (V, yl = {a,j | i € /}) and V = (Pv \ v E V) gives a tree representation of the laminar family T = (Xj \ i € / ) . If we replace an arc a^ by series arcs a^i and a{l" with a vertex w^/ between them such that d+a,i1 = d+ail», d~cii1 = d~ail' and d+ail' = d~a,i1", and if we define Pv. , = 0, then the pair of the augmented tree T' = (VU {Vil,}, {A - {ah}) U {ah,, ahn}) a n d V = (PVi \ie(I{h}) U {io,H',h"}) represents the laminar family T' = (X{ \ i € (/ — {ii}) U {io^i'^i"}) with repeated members Xixi = X^//. Repeated members can be treated in this way. We thus have Lemma 3.48: A family of subsets of a finite set E is laminar if and only if it has a tree representation such that the tree is a directed tree toward the root. Next, let T = (Xi \ i € /) be a cross-free family of subsets of E, i.e., for each Xi, Xj (i,j € /) we have at least one of the following four: (1) Xi n X3 = 0, (2) X% C Xj, (3) X% D Xj and (4) X% U Xj = E. Choose an arbitrary element eo G E and define f=(Xi\

iel),

(3.170)

where, putting IQ = {i \ i G I, eo 6 Xj},

v- _ J E~xi ^"jx,

if*e Jo, ifteJ-Jo

m ? n ( 3 7 )

for each i E I. We can easily see that J- is a laminar family, since J- is cross-free and Xi U Xj C E — {eo} for any i,j G / . Therefore, T can be represented by a pair of a tree T = (V, A) and a family V = (Pv \ v EV), where A = {Sj | i G / } . Construct a tree T = (V, A) by reorienting the arcs

3.4. Intersecting- and Crossing-Submodular Functions

89

en (i € /o) of tree f = (V, A). Then, the pair of the tree T and the family V represents the original cross-free family T. Now, we have L e m m a 3.49 [Edm+Giles77]: A family of subsets of E is cross-free if and only if it has a tree representation. The following lemma, concerning the decomposition of uniform crossfree families, is fundamental. L e m m a 3.50: Let Q = (Xi \ i £ I) be a cross-free family of subsets of E which uniformly covers each element of E, i.e., Ve e E: \{i\i€l,

e E Xi}\ = const. > 0.

(3.172)

Then, Q is the direct sum of families each of which forms a partition or a copartition of E. (Proof) Since {E} is a partition and {0} is a copartition of E, we suppose that $,E £ G and that Q / 0. From Lemma 3.49, the cross-free family Q = (Xi | i G /) can be represented by a pair of a tree T = (V, A), with a vertex set V and an arc set A = {a-i \ i € / } , and a family V = (Pv\veV)

(3.173)

of subsets of E, where nonempty P^'s form a partition of E. Note that by the assumption we have Pv + 0 (3.174) for any leaf v of T. Now, from the assumption and (3.174) there exists at least one Xi such that Xi (£ {0, E}. Therefore, there exist distinct vertices v\ and V2 of T satisfying the conditions that PV1 + 0,

Pv2 + 0

(3.175)

and that for any vertex u ^ {^1,^2} lying on the unique path from v\ to V2, denoted by Q(v\, V2), in T we have Pu = 0.

(3.176)

It follows from (3.172) that the number of positively oriented arcs in Q(v\,V2) is equal to the number of negatively oriented arcs in Q(v\,V2)- (If this is

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

90

not the case, the value of (3.172) for any e\ £ PVl can not be equal to that for any e% E Py2-) I n particular, there exists at least one positively oriented arc and at least one negatively oriented arc in Q(v\,V2)- Hence, there is at least one vertex u $_ {^1,^2} on Q(v\,V2). Let u be the vertex on Q(vi,V2) adjacent to v\ and let {i>2, ^3, , Vk\ (k > 2) be the maximal set of vertices of T such that for each I = 2, 3, , k, (i) PVl 7^ 0, (ii) the vertex u lies on the unique path Q(vi,vi) from v\ to v\ in T and (iii) any vertex u $. {v\,vi} lying on Q(v-\_,vi) satisfies (3.176). Moreover, let cij^ £ A be the arc connecting v\ with u and for each I = 2,3, , k let a^j) € A be the arc in Q(v\, v{) such that the orientation of the arc a,j^ is opposite to the orientation of the arc &j(i) and any arc a, (^ aj(i)) between a,j(i) a n ( i a,j(j[) in Q(vi,vi) has the same orientation as Oj(i). (Note that arcs o^) (I = 2, 3, , k) are not necessarily distinct.) (See Fig. 3.6.)

Figure 3.6: A tree representation. Let us define n = {Xm

I XJ{1)

GG,l
Then, (i) if 5 + aj(!) = v\, II is a partition of i? and (ii) if d~a,j(i^ = v\, II is a copartition of i?.

(3.177)

3.4. Intersecting- and Crossing-Submodular Functions

91

Therefore, the family Q contains a subfamily which forms a partition or copartition of E. Define g' = (xt\iei-

{j(i) 11 = 1,2,

, k}).

(3.178)

If Q' is not empty, then Q' also satisfies (3.172) and is cross-free. Repeating the above-mentioned argument, we see that Q is the direct sum of families each of which forms a partition or a copartition of E. Q.E.D. (b) Crossing-submodular functions Let J- be a crossing family of subsets of a finite set E with 0, E € J- and / a crossing-submodular function on T with /(0) = 0 (see Section 2.3 for the terminology). We assume that R is the set of reals (or rationals). Define

P(/) = {* | x G KE, VX G T: x(X) < f(X)}.

(3.179)

Note that P(/) is nonempty. Now, we would like to find the tightest system of inequalities with {0,1}coefficients which represents the polyhedron P(/). Therefore, define f(Y) = max{x-(y) | x e P(/)}

(3.180)

for any Y C E, where we put f(Y) = +CXD if the value of x(Y) can be made arbitrarily large for x G P(/)- Note that by definition (3.180) we have /(0) = 0. By the linear programming duality theorem (Theorem 1.10) we have

f(Y) = mm{J2 f(X)c(X) xer

\ (3.182), (3.183)},

(3.181)

where

£

c{X) = xr(e) = {l

\Hy\

c(X)>0 ( l e f ) .

(e G E),

(3.182) (3.183)

We have f(Y) = +oo if and only if there is no such c(X) (X G T) that (3.182) and (3.183) are satisfied. We thus have a set function / : 2E RU{+oo}.

92

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

Since the minimum value of (3.181) can be attained by rational c(X) (X E J7), (3.181)~(3.183) can be rewritten as follows.

f(Y) =

| (3.185)-(3.187)},

i)

(3.184)

where with I.Gf,

XtCY

(i € / )

(3.186)

and |{i | i € / , e e X J I =const. = /x(^,y) > 0

(e € Y).

(3.187)

Note that Q = {Xi \ i € / ) is a family of subsets Xi of £J, so that we may have Xi = Xj for distinct i,j € / . (3.187) means that the family Q uniformly covers each element of Y. By the definition of the set function / : 2 E ^ R U {+00} we have f(X)
(3.188) (3.189)

If we decrease the value of any f(X) (X C E), (3.189) does not hold. Since we have not used the crossing-submodularity of / in the above argument, (3.179)^(3.189) are valid for any set function on any family of subsets of E. In the following, we simplify (3.184)~(3.187) by use of the crossingsubmodularity of / . From the crossing-submodularity of / and (3.184)~(3.187), we can restrict admissible families Q in (3.184)~(3.187) to those which satisfy (i) Q is a cross-free family,

(3.190)

(ii) 0 i G,

(3.191)

to attain the minimum of (3.184). (For, let Q be a family which satisfies (3.185)~(3.187) and attains the minimum of (3.184). If there exists a crossing pair of Xi and Xj in Q, then put Xi <- Xi U Xj,

X3 <- Xi n Xj.

(3.192)

3.4. Intersecting- and Crossing-Submodular Functions

93

After this replacement Q = (Xi | i € /) remains to satisfy (3.185)~(3.187), while the sequence of the values of \X{\ (i € /) arranged in order of nonincreasing magnitude becomes lexicographically greater than the previous one. Because of the finiteness character we can repeat the replacement (3.192) for crossing pairs in Q only finitely many times and eventually we obtain a cross-free family Q which satisfies (3.185)^(3.187) and attains the minimum of (3.184). Furthermore, if Q contains the empty set, it is redundant and can be removed.) Theorem 3.51: Let fp(E) and fq(E) be defined by fp(E) = m i n { ^ f(Xi)

{Xi \ i € / } : a partition of E,

iei

ViG /: Xi e^},

fq(E) = mini —

^ f(Xj)

(3.193)

{Xj \ j € J } : a copartition of E, Vj € J: X3 € F, \J\ > 3}. (3.194)

Then we have

f(E) = min{/p(£0, fq(E)}.

(3.195)

(Proof) We can restrict admissible families Q = (Xi | i € /) in (3.185)^(3.187) with Y = E to those further satisfying (3.190) and (3.191). For such a family Q, from Lemma 3.50, Q is the direct sum of partitions and copartitions of E. Since every subfamily of Q forming a partition or a copartition of E satisfies (3.185)^(3.187) with Y = E, (3.190) and (3.191), it follows that we can restrict admissible families Q = (Xi | % E I) to those which form partitions and copartitions of E with X^ E T (i E I). Note that the copartition {0} of cardinality 1 is excluded by (3.191) and that copartitions of cardinality 2 are also partitions. We thus have (3.193)^(3.195). Q.E.D. From Theorem 3.51 we can show Theorem 3.52: The following four statements are equivalent:

94

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

(i) The polyhedron defined as the intersection of P(/) and the hyperplane x{E) = f(E), i.e., B(/) = {x | x € RE, VX € ^ : x(X) < /(X), x(E) = / ( £ ) } , (3.196) is nonempty. (ii) f(E) = f(E).

(3.197)

(iii) / ( £ ) = / p ( £ ) = (/#)p(£),

(3.198)

where (f*)p(E) = m a x { ^ f*(Yj)

\ {Yj \ j £ J}: a partition of E, Vj € J: E-Yj EJ7}.

(3.199)

(iv) For any partitions {JQ \ i £ 1} and {Yj | j £ J} of E with Xi £ T (i E I) and E - Yj € F {j € J) we have

E/ # (^)
(3-20°)

iei

(Proof) The equivalence between (i) and (ii) follows from the definition of f(E). To show the equivalence between (ii) and (iii), recall (3.193)~(3.195). If f(E) = f(E), we have fp(E) > f(E), while from (3.193) we have fp(E) < f(E) since E € T. We thus have fp(E) = f(E). Moreover, since 0 € J7, we have from (3.199) (f*UE)>f#(E) = f(E).

(3.201)

If f(E) = f(E), we have from (3.194) and (3.195) f(E) < fq(E), i.e., (|j|-i)/(£;)<E/(^)

( 3 - 202 )

for any copartition {Xj \ j £ J} of E such that Xj £ T (j £ J) and \J\ > 3. Note that (3.202) holds for | J\ = 1 and 2, due to the assumption that f(E) < fp(E). (3.202) is rewritten as £/#(^)
(3.203)

3.4. Intersecting- and Crossing-Submodular Functions

95

for any partition {Yj \ j £ J} of E such that E — Yj € JF (j G J). Hence, (f*)p(E) < f{E). Consequently, from (3.201) we have (f*)p(E) = f(E). Conversely, suppose that f(E) = fp(E) = (f*)p(E). Then we have (3.203) and hence f(E) < fq(E). Therefore, f(E) = f(E). Finally, we show the equivalence between (iii) and (iv). Since we always have

fp(E) < f(E) < (f*)p(E), (iv) implies (iii). The converse, (iii) = > (iv), is immediate.

(3.204) Q.E.D.

Theorem 2.6 follows from Theorem 3.52. For proper subsets of E the value of / is expressed as follows.

Theorem 3.53: For each nonempty proper subset Y of E, f(Y) = m i n j ^ f(Xi)

{Xi \ i e / } : a partition of Y, Vi € /: Xi e f}.

(3.205)

(Proof) By the same argument as in the case of Y = E in the proofs of Lemma 3.50 and Theorem 3.51, we see that we can restrict admissible families Q in (3.185)~(3.187) to those which form partitions and copartitions of Y. Let V* = {Xi | % € / } be a copartition of Y with X{ € T (i G /) and |/| > 3. (Note that if |/| = 2, V* is also a partition of Y.) Then, for any distinct Xi, Xj € V*, Xi and Xj cross. It follows from (3.190) that we need not consider copartitions of Y. Q.E.D. Note that / is the Dilworth truncation of / when B(/) is nonempty. We also have Theorem 3.54: For any X, Y C E with XUY ^ E, if f(X), f(Y) < +oo, then f(X) + f(Y) > f(X U Y) + f(X n Y).

(3.206)

(Proof) If XHY = 0, (3.206) is immediate. Therefore, suppose that X,Y
96

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

partition {Xi | i £ / } of X and some partition {Yj | j £ J } of Y such that Xi £ .T7 (i £ / ) and Y,- £ F (j £ J), we have

/(x) + /(y) = E / ( * ) + E /(**) ie/

(3-207)

jeJ

due to Theorem 3.53. Let Q = (Z& | k £ / © J ) be the direct sum of families (Xi \ i E I) and (Yj \ j E J). Since X U Y ^ E, one of the following three statements holds for any Z^.Zj £ Q: (i) Zi and Zj are disjoint,

(3.208)

(ii) Zi C Z,- or Z,- C Zi}

(3.209)

(iii) Zj and Zj cross.

(3.210)

If Zj and Zj cross, replace Zj and Zj as Zi^-ZiUZj,

Zj <- ZiHZj.

(3.211)

Repeat this replacement until the current family Q = (Z& | k E I © J ) becomes a cross-free family and satisfies (3.208) and (3.209) for any Z,, Zj £ Q. By the same argument made after (3.192), this replacement process terminates in a finite number of steps. The finally obtained Q is the direct sum of two families which form a partition {Xi \ i £ 1} (Xi £ T (i £ I)) of X U Y and a partition {Yj \ j £ J} (Yj £ T (j £ J)) of X D Y. Since the replacement (3.211) reduces the value of (3.207), we get

f(X) + f(Y) > X/(^) + E/(^) iei

jeJ

> f(XUY) + f(XnY).

(3.212) Q.E.D.

It follows from Theorem 3.54 that the family J- defined by P ={X \X C\E, f(X) < +oc}

(3.213)

is a cointersecting family (i.e., X,Y £p, XUY / E => XUY, X(~)Y £ P) and that / restricted to JF is a cointersecting-submodular function on P (i.e., f(X) + f(Y) > f(X U Y) + f(X n Y) for any I . ^ e f with XUY^E).

3.4. Intersecting- and Crossing-Submodular Functions

97

Now, let us consider the polyhedron

B(/) = {x | x € nE, VX € T: x(X) < f(X), x(E) = f(E)}

(3.214)

which is the intersection of P(/) and the hyperplane x(E) = f(E), and suppose that B(/) is not empty (see Theorem 3.52). We examine the structure of B(/). For each proper subset Y c E, define h(Y) = max{x(Y) | x € B(/)}.

(3.215)

Then, by the linear programming duality theorem, MY) = min{ Y, f{X)c{X)

\ (3.217), (3.218)},

(3.216)

where Y,

c(X) =XY(e)

c(X) > 0

(eEE),

(3.217)

(lef.I^B).

(3.218)

Similarly as (3.184), (3.216) can be rewritten as

b(Y)

=mi"{,(g,Y)-l{g,E-Y) [ W - "^E - ^ 1 (3.220)-(3.224)},

(3.219)

where g = (Xi\i£l), Xi^T, \{i\eeXi,

X% + E

(3.220) (iel),

iel}\ = const. = /x(^, Y)

|{i | eGXi, ie I}\ = const. = fj,(g,E-Y)

(3.221) (e € y )

(3.222)

(e£E-Y),

(3.223)

fjL(g,Y)>n(g,E-Y). (3.224) It should be noted that (3.219) is valid for any set function / . By use of the set function / : 2 £ ^ R U {+00} defined by (3.184) (or (3.195) and (3.205)), the polyhedron B(/) of (3.214) can also be expressed as

B(/) = {x I x € KE, \/X € T: x(X) < f(X), x(E) = f(E)(=

f(E))}, (3.225)

98

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

where T is defined by (3.213). Therefore, we can replace / and T in (3.219) and (3.221) by / and J-, respectively. For Y c E, if {E-Xi \ % G 7} is a partition of E-Y, we call {X{ | i G 7} a co-partition of E — Y augmented by Y. Theorem 3.55: For each nonempty Y C E,

/ 2 (y ) = m i n{^/(X l )-(|7|

-l)f{E)

iei

{E — Xi | i G 7}: a partition of E — Y, ViG 7: XiGfi}.

(3.226)

(Proof) If / and T in (3.219) and (3.221) are replaced by / and f, we can restrict admissible families Q in (3.220)~(3.224) to those which satisfy (3.220)~ (3.224) (with T replaced by T in (3.221)) and the following (i)~(iv): (i) Q is a cross-free family,

(3.227)

(ii) for any Xu Xj € Q we have Xt n Xj ^ 0,

(3.228)

(hi) Q does not contain a subfamily which forms a copartition of E, (3.229) (iv) E <£ Q.

(3.230)

(Here, (i)~(iii) follow from Theorems 3.51, 3.53 and 3.54. (iv) follows from the form of (3.219).) From (i), the family Q = [X{ \ i G I) can be represented by a pair of a tree T = (V, A) and a family V = {Pv\veV),

(3.231)

where A = {aj | i € 7} and nonempty P^'s form a partition of E as in Lemma 3.49. From (ii), T is a directed tree. (For, if there were distinct arcs a,j and cij in T such that d~'a\ = d~cij, we would have Xi D Xj = 0.) Let VQ be the root of T. If PVo = 0, then Q contains a subfamily which form a copartition of E. Therefore, PVo ^ 0 due to (iii). Since for each e £ E the number of i's for which e € Xi should be taken from the fixed set of two distinct values of (3.222) and (3.223), for any leaf u of T every vertex w ^ {u, vo} lying on the unique path Q(vo,u) connecting vo with u in T gives Pw = 0, (3.232)

3.4. Intersecting- and Crossing-Submodular Functions

99

where note that Pu ^ 0 due to (iv). It follows from (3.224) that Pvo = Y

(3.233)

and that d+at = v0}

{E-Xi\iel,

(3.234)

is a partition of E — Y. Let Q1 be the family obtained from Q by deleting X^s such that d+a{ = VQ. If Q' / 0, then Q< also satisfies (3.221)~(3.224) (with T replaced by P) and (3.227)~(3.230). (The tree representation of Q' with root VQ is obtained by contracting or shortcircuiting arcs a-i such that d+ai = VQ in T.) It follows that Q is a direct sum of families which form copartitions of E — Y augmented by Y. Since any family which forms a copartition of E — Y augmented by Y also satisfies the conditions required for Q, the minimum of (3.219) is attained by a family Q which forms a copartition of E-Y augmented by Y. Q.E.D. We define f2(E) = f(E)(=f(E)).

(3.235)

We now have a set function J2 from 2E to R U {+oo}. Theorem 3.56: For any X, Y C E such that / 2 (X), f2(Y) < +oo, we have MX) + MY) >f2{XUY)

+ f2(X n Y).

(3.236)

(Proof) If X G {0,E} or Y G { 0 , ^ } , then (3.236) is trivial. So, suppose X $_ {0, E}, Y (£ {0, E} and X, Y E P. Then, from Theorem 3.55, for some partition {E — X{ \ i G / } of E — X and some partition {E — Yj \ j G J }

of E-Y

with Xi E P (i € /) and Yj € P (j E J), we have

;2po + ;2(Y) = ^ / ( ^ - ( i / i - i ) / ^ ) i€l

+ Y,fiYj)-(\J\-l)f(E).

(3.237)

Let Q = (Zfc | k G / © J ) be the direct sum of families (Xi \ i G / ) and (Yj | j G J ) . If, for Zj, Z,- G g, we have (E - Z,{) D (E - Zj) / 0, ZiC\(E- Zj) ^ 0 and (E - Z{) n Z, ^ 0, then replace Z{ <- Zi U Z,-,

Zj <- Zi n Zj.

(3.238)

100

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

Repeat such a replacement until there is no such pair of Z\ and Zj in Q. This process terminates in finitely many steps. We can easily see that the finally obtained Q is the direct sum of families Gi = (X* \ i € /*) and Q2 = (Y* | j € J*) such t h a t {E-X*

\i€ I*} is a partition of

E-(XUY)

and {E - Y* \ j € J*} is a partition of E - (X n Y), where, if X n y = 0, the family C?2 may be composed of the empty set alone. For any pair of Z{ and Zj to be replaced by (3.238) we have ZiUZj ^ E. It follows from Theorem 3.54 and (3.237) that

h(x) + h(Y) > Y,f(xD-(\n-Vf(E) iei*

+ Y^f(Y*)-(\r\-i)f(E) jeJ*

> f2(XUY) + f2(XnY).

(3.239) Q.E.D.

From Theorem 3.56, V2 = {X\XCE,

f2(X) < +00}

(3.240)

is a distributive lattice with set union and intersection as the lattice operations, join and meet. Denote by f2 the function obtained by restricting the domain 2E of f2 to V2. Then, (T>2, f2) is a submodular system on E and the polyhedron B(/) denned by (3.225) is also expressed as B(/) = {x\xGllE,VXe

V2: x{X) < f2(X), x{E) = f2{E){= f(E))} (3.241) which is the base polyhedron B(f2) associated with the submodular system (V2J2). Since different submodular systems give different nonempty base polyhedra, nonempty B(/) uniquely determines a submodular system (V2, f2) such that B(/) = B(/ 2 ). Moreover, we see from (3.235) and Theorems 3.53 and 3.55 that if B(/) is nonempty, each f2(X) (X € T>2) is expressed as a linear combination of / ( y ) ' s (Y € T) with integer coefficients. Therefore, if / is integer-valued, so is f2. This completes the proof of (ii) of Theorem 2.5.

3.4. Intersecting- and Crossing-Submodular Functions

101

It should be noted that (3.226) can be rewritten in a dual form as ff{Z)

= m a x { ^ f*(Yj)

| {Yj \ j € J } : a partition of Z, Vj € J: E-Yj£

T)

(3.242)

for each nonempty proper subset Z C E. Here, f% is the Dilworth truncation of the intersecting-supermodular function / # on P. (c) Intersecting-submodular functions Let / : JF —> R be an intersecting-submodular function on an intersecting family .F. We assume that /(0) = 0 if 0 € T. Consider the polyhedron P(/) = {x | x

e

RE, VX € ^

x(X) < / ( X ) } .

(3.243)

Since intersecting-submodular functions are crossing-submodular, P(/) is expressed as P(/) = {x\x£RE,

yX C E: x(X) < f(X)},

(3.244)

where / is denned by (3.193)~(3.195) and (3.205). Since T is an intersecting family and / : T —> R is intersecting-submodular, we can restrict Q in (3.184)~(3.187) to those which satisfy (3.186), (3.187), (3.190), (3.191) and the following: (iii) for any Xi,Xj Xj C Xi,

€ Q such that X\ Pi Xj ^ 0, we have Xi C Xj or (3.245)

i.e., Q is a laminar family. (3.193) and (3.194) satisfy

In particular, fp(E) and fq(E) defined by

fp(E) < fq(E).

(3.246)

It follows from (3.246) and Theorems 3.51 and 3.53 that for any nonempty Y
| {Xi \ i £ I}: a partition of Y, Vi € / : Xi € T},

(3.247)

102

II. SUBMODULAR SYSTEMS AND BASE

POLYHEDRA

and we have /(0) = 0 by definition (3.180). Function / is the Dilworth truncation of / . Theorem 3.57: Let f: JF —> R be an intersecting-submodular function. For any X,Y C E and f: 2 £ ^ R U {+00} defined by (3.247), iff(X),f(Y) < +00, then f{X) + f(Y)

> f(X U Y) + f(X n Y).

(3.248)

(Proof) Since / is an intersecting-submodular function on JF, Theorem 3.54 holds for X,Y C E with X UY = E as well. Q.E.D. Define Vx = {X I X C E, f(X)

< +00}

(3.249)

and let f\ be the restriction of / to T>\. Then, from Theorem 3.57, T>\ is a distributive lattice and f\ is a submodular function on T>\. Since polyhedron P ( / ) in (3.243) is expressed in terms of /1 as P(/) = {x I x e HE, VX € Pi: x(X) < h(X)} which is the submodular polyhedron P(/i) associated system ( P i , / i ) on E. Since different submodular systems give different lar polyhedra, the intersecting-submodular function determines the submodular system {V\, f\) such that over, from (3.247), /1 is integer-valued if / is. This completes the proof of (i) of Theorem 2.5.

(3.250)

with the submodular associated submodu/ : J- —> R uniquely P ( / ) = P(/i)- More-

3.5. Related Polyhedra We show some polyhedra which are closely related to base polyhedra and submodular/supermodular polyhedra. (a) Generalized polymatroids A. Frank [FrankSlb] introduced the concept of generalized polymatroid. Suppose that a submodular system (T>i,f) and a supermodular system (T>2,g') on E' satisfy VX eVl,VY£V2:

X - Y G V1} Y - X G V2, f'{X)

- g'(Y) > f{X

-Y)-

g'{Y - X). (3.251)

103

3.5. Related Polyhedra

[Here, the unique maximal elements of V\ and T>2 can be proper subsets of E', while we define submodular and supermodular polyhedra by systems of inequalities for V\ and V2 as in the original definitions.] Then the polyhedron P(f',g') defined by P(f',g') = {x\xeRE',VX

GV1:x(X) < f'(X), Vy e V2: x(Y) > g'(Y)}

(3.252)

is called a generalized polymatroid or g-polymatroid. (The same polyhedron is also considered by R. Hassin [Hassin78,82] for the case where V\ = V2 = 2E'.)

Generalized polymatroids are characterized by the following theorem, which is implicit in [Frank8fb] (also see [Schrijver84a]) (see Fig. 3.7).

Figure 3.7: A generalized polymatroid as a projection of a base polyhedron. Theorem 3.58 [Fuji84a]: For the base polyhedron B(/) associated with a submodular system (V, f) on E, the projection ofB(f) along an axis e € E on the hyperplane x(e) — 0 is a generalized polymatroid P(f',g') in KE

104

II. SUBMODULAR

SYSTEMS AND BASE

POLYHEDRA

with E' = E — {e}, where Vl V2

= {X | e £ X € V}, = {E-X \eeX eV},

(3.253) (3.254)

/ ' is the restriction of f to V>\, and g' is the restriction of f^ to T>2Conversely, every generalized polymatroid in HE is obtained in this way. For each generalized polymatroid P(f'.g') with T>i C 2 £ (i = 1,2) such a base polyhedron B ( / ) in R, with E = E' U {e} is unique up to translation along the new axis e, and the two polyhedra P(/',
(X e V),

x(E) = f(E).

(3.255) (3.256)

Choose an element e 6 E. From (3.256) we have x(e) = f(E)-x(E-{e}).

(3.257)

Substituting (3.257) into (3.255), we have VX € V with e £ X: x(X) < f(X), V l e P with e e X: x{E - X) > f(E) - f(X).

(3.258) (3.259)

(3.258) and (3.259) is rewritten as VXePi: x(X) < f'(X),

(3.260)

WY E V2: x(Y) > g'(Y), (3.261) where V\ and T>2 are, respectively, defined by (3.253) and (3.254), / ' is the restriction of / to T>\ and g' is the restriction of / * to T>2- It follows from (3.260) and (3.261) that the projection of B(/) along the axis e on the hyperplane x(e) = 0 is the generalized polymatroid ~P(f',g') in HE with E' = E — {e}. Note that (3.251) follows from the submodularity of / . Now, we show the converse. For an arbitrary generalized polymatroid / P(/ ,<7/) in R with a submodular system (Z?i,/') and a supermodular system (T>2,g') on E' let e be a new element not in E' and define E = E'U {e},

(3.262)

3.5. Related Polyhedra

105 V20 = {E-X \ X eV2},

(3.263)

V = V1UV2°.

(3.264)

Then, T> is a distributive lattice with 0, E € P due to (3.251). Also, define a function / : P —> R by f(X) f(X)

= f'(X)

(X € Pi),

= f(E)-g'(E-X)

(XeV~2°),

f(E) = c,

(3.265) (3.266) (3.267)

where c € R is arbitrary but fixed. From (3.251), (3.265) and (3.266) we have for any X € P i and Y E P2 f(X) + f(Y) = f'(X) +

f(E)-g'(E-Y)

+ f(E)-g'((E-Y)-X) > f'(X-(E-Y)) f = f(XnY) + f(E)-g (E-(XUY)) = f(XDY) + f(XUY). (3.268) Therefore, / : T> —> R is a submodular function on the distributive lattice V. Moreover, for some a € R, (x, a) € B(/) if and only if VX € Pi: x(X) < f(X) = f'(X), VX € V20: x(X - {e}) + a < f(X) = f(E) - g'(E - X), x(E') + a = f(E).

(3.269) (3.270) (3.271)

Eliminating a in (3.270) by using (3.271), we have VX € V~2°: x{E -X)> g'{E - X)

(3.272)

or Vy € V2: x(Y) > g'{Y).

(3.273)

Therefore, for the submodular system (£>,/) defined by (3.262)^(3.267) P ( / , g') = {x\xe KE, 3a € R: (x, a) € B ( / ) } .

(3.274)

Moreover, it is clear that P ( / ' ,
106

II. SUBMODULAR SYSTEMS AND BASE

POLYHEDRA

It follows from Theorem 3.58 that extreme points, extreme rays, faces etc. of P(/', Z + is the rank function of a matroid M i on E', g': 2E —> Z+ is the dual supermodular function of the rank function of a matroid M2 on E' and the pair of / ' and g' defines a generalized polymatroid, we call the ordered pair (Mi,M2) a strong map. This definition of strong map is equivalent to the ordinary one (see, e.g., [Welsh76]). If (Mi, M2) is a strong map and the rank of M i is equal to the rank of M2 plus one, the submodular system (T>, / ) on E = E' U {e} appearing in Theorem 3.58 is a matroid M 3 such that M 3 - {e} = M i and M 3 /{e} = M 2 , if we define f{E) = f'(E'). Frank [Frank81b] originally defined generalized polymatroids in terms of intersecting families. This corresponds to the fact that a crossingsubmodular function on a crossing family determines the base polyhedron associated with a submodular system (see Theorem 2.5). Note that if V C 2E is a crossing family, then T>i (i = 1,2) defined by (3.253) and (3.254) are intersecting families. (For more details on generalized polymatroids, see [Frank+Tardos88].)

(b) Polypseudomatroids The concept of pseudomatroid was introduced by R. Chandrasekaran and S. N. Kabadi [Chandrasekaran+KabadiSS]. The same or similar concepts were independently considered by A. Bouchet [Bouchet87] as A-matroid, A. Dress and T. F. Havel [Dress+Havel86] as metroid, M. Nakamura [Nakamura88b] as universal polymatroid and L. Qi [Qi88] as ditroid. We shall consider (poly-)pseudomatroids of Chandrasekaran and Kabadi from the point of view of submodular systems. Denote by 3E the set of all the ordered pairs (X, Y) of disjoint subsets r X{x,Y) € of E. [An element (X, Y) of 3E is identified with a {0, s e = e = r r R defined by X(X,Y)( ) 1 f° e E X, X(X,Y){ ) ~1 f° e € Y and X(x.Y){e) = 0 f° r e € E — (X U Y). Such an ordered pair (X, Y) is called a signed set] Let / : 3E —> R be a function with /(0, 0) = 0 such that for

each {Xi,Yi)e3E f(Xu

{i = 1,2)

Yi) + f(X2, y 2 ) > /((Xi U X2) - (Yi U Y2), (Yi U Y2) - (Xx U X2))

+f(x1nx2,Y1nY2).

(3.275)

3.5. Related Polyhedra

107

Define a polyhedron P,(/) = {x | x G R^, V ( I 7 ) € 3 £ : x(X) - x(F) < /(X,F)}. (3.276) The polyhedron P*(/) is called a polypseudomatroid ([Chandrasekaran+ Kabadi88], [Kabadi + Chandrasekaran90]) and / its rank function (see Fig. 3.8). Polyhedral studies are also made in [Nakamura88b] and [Qi88,89]. A pseudomatroid is a set-theoretical version of a polypseudomatroid. [Since f(X,Y) is submodular both in X with any fixed Y and in Y with any fixed X, f is called a bisubmodular function (like 'bilinear' in a bilinear form) and a polypseudomatroid a bisubmodular polyhedron later in [Bouchet+Cunningham95], [Ando+Fuji96], [Fuji+Patkar95], etc. (Note that f(X,Y) is not submodular in (X,Y) in general as a bilinear form is not linear.) Also see [Borovik+Gelfand+White03] for related topics in Coxeter matroids.] x(2)

Figure 3.8: A polypseudomatroid. We call a pair (S, T) G 3E such that SUT = E an orthant of KE. For each orthant (S, T) denote by 2^ T ) the set of all the pairs (X, Y) such that X C S and Y CT.

108

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

Now, choose an orthant (S, T). Then for each (JQ, Yj) G 2(-s^ (i = 1,2) we have from (3.275) f(Xu

Yi) + f{X2, Y2) > f{XlUX2,

Yl\JY2) + f{X1C\X2,Yl

n y 2 ). (3.277)

This means that for the orthant (5, T) the function / ' : 2 £ —> R defined by /'(I)=/(5nl,Tnl)

(A"C£)

(3.278)

is a submodular function on 2E. Define P(S,T)(/) = {x\xeRE,

V ( X Y ) G 2(S'T):x-(X) -x-(F) <

f(X,Y)}. (3.279) The polyhedron P(S,T) (/) is expressed by the submodular polyhedron P ( / ' ) associated with the submodular system (2E', / ' ) as follows. P(S,T)(/) = {x I y G P(/')> Ve G 5:x(e) = j/(e),Ve G T:x(e) = -y(e)}. (3.280) Therefore, the combinatorial properties of P(,s,T){f) a r e the same as those of P ( / ' ) and the greedy algorithm described in Section 3.2.b works for P(ST)(/) mutatis mutandis as for P ( / ' ) . Define B(S,T)(/)

= {x | x G P ( S ,T)(/), x(5) - x ( T ) = / ( 5 , T ) } .

(3.281)

We call B( S T )(/) the base polyhedron in the orthant (5, T) of the polypseudomatroid P*(/). When f(S,T) = 0 and B ( S i T ) (/) C R^, B ( S > T ) (/) is the set of linked vectors of a polylinking system (or a polybimatroid); a set theoretical version of the system is called a linking system (or a bimatroid) (see [Schrijver78,79] and [Kung78]). From Theorem 3.22 we have Corollary 3.59: A vector x G R is an extreme point of B(sT)(f) (3.281) if and only if for a maximal chain C: $ = A0dAl(Z---(lAn

=E

J12

(3.282)

of 2E we have

f(snAuTn Ai) - f(sn A^,Tn

A^)

_ / x(Ai - Ai-i)

if Ai - Ai-i C S

"

if A - A^ C r

-x{Ai - A-i)

,

s

^ ' ^ ^

3.5. Related Polyhedra for each i = 1, 2,

109

, n.

We also have Lemma 3.60: For each orthant (S, T) of RE we have B M (/)CP,(/).

(3.284)

(Proof) Suppose x € ~B(s,T)(f)- Then, for any (X,Y) € 3^ we have from (3.275)

x(X)-x(Y)-f(X,Y) = x{X) - x(Y) - f(X, Y) + x(S) - x{T) - f(S, T) < x(S-Y)-x(T-X)-f(S-Y,T-X) +x(s nx)- X(T n Y) - f(s n x, T n Y) < 0.

(3.285)

Hence x € P*(/).

Q.E.D.

We see from Lemma 3.60 that B(s,T)(f) f° r each orthant (S, T) is a face ofP*(/). Since P*(/) = ni P (5,T)(/) I (S,T): an orthant of KE},

(3.286)

for any w. E ^ R w e have from Lemma 3.60 m a x { E w(e)x(e) \ x e P(s,T)(/)} eG-B

> max{^™(e)i(e) | x e P*(/)} eeE

> max{^ W ( e )x(e) | i £ B w ( / ) } .

(3.287)

ee_B

For an orthant (S,T) such that w(e) > 0 ( e £ S) and w(e) < 0 (e G T), (3.287) holds with equality. Therefore, the problem of maximizing the linear function Y^eeE w(e)x(e) over the polypseudomatroid P*(/) is solved by maximizing the same linear function over P(s.T)(/) o r ^(S,T){f)- Hence the greedy algorithm as in Corollary 3.59 works for the polypseudomatroid

110

II. SUBMODULAR SYSTEMS AND BASE

POLYHEDRA

P*(/) and the union of all the extreme points of B(s jT )(/) for all the orthants (S, T) is exactly the set of all the extreme points of P*(/). Also, we see from Lemma 3.60 (and Lemma 3.2) that for each (X, Y) G 3E / ( X Y) = max{x(X) - x(Y) | x G P*(/)},

(3.288)

so that / is uniquely determined by P*(/). The greedy algorithm is given as follows. Here, note that we consider a maximization problem rather than a minimization problem (cf. Section 3.2.b). Greedy algorithm for polypseudomatroids Step 1: Find a sequence (ei, e,2, e n ) of all the elements of E such that w e ( i ) | > |tt;(e2)| > > |i^(en)|- Choose an orthant (S,T) such that w(e) > 0 (e G S) and w(e) < 0 (e G T). Step 2: Let A{ be the set of the first i elements of the sequence (ei, e2, , en) for each i = 1, 2, , n and compute a vector x G HE by (3.283). (x is a maximizer of the linear function J^eG-E w(e)x(e) over the polypseudomatroid P*(f).) (End) Theorem 3.61 [Chandrasekaran+Kabadi88], [Nakamura88b]: For any function f: 3E — R define P»(/) = {x | x G R £ , V(X,y) G 3 £ : x ( X ) - x(Y) < f(X,Y)}.

(3.289)

The greedy algorithm of the type described above works for P*(/) if and only if f is the rank function of a polypseudomatroid. (Proof) It suffices to show the "only if" part. Suppose that the greedy algorithm described above works for P*(/). Then, for any (Xi,Yi) G 3E (i = 1,2) choose a maximal chain C of (3.282) containing [X\ n X2) U (Yi n Y"2) and {{X1 U X 2 ) - (Y1 U y 2 )) U ((Yi U y 2 ) - (Xi U X 2 )) in it and define the vector x G R £ by (3.283), where (S.T) is an orthant such that (XL U X2) ~ (Yi U y 2 ) C S and (Yi U Y2) - (Xi U X2) Q T. From the assumption we have x G P*(/). It follows from the definition of x that x(Xi) - x(Yi) < f(Xu Yt)

(1 = 1,2),

(3.290)

x((Xi u x2) - (Yx u y2)) - x((Y! u y2) - {xx u x2)) = / ( ( X i U X 2 ) - (Yx U y 2 ), (Yi U Y2) - (Xi U X 2 )),

(3.291)

3.5. Related Polyhedra

111

x{xl n x2) - x{Yl n y2) = f{xl n x 2 , yx n y 2 ).

(3.292)

From (3.290)~(3.292), /(Xi,yi) + / ( x 2 ) y 2 )

> x(x 1 )-x(y 1 ) + x(x 2 )-x(y 2 ) = x{{xl u x 2 ) - (n u y2)) - x((yx u y2) - (xx u x 2 )) +x(xx n x 2 ) - x(yx n y2) = f{(x1 u x2) - (Y1 u y2), {Y1 u y2) - (xx u x2)) +f(x1nx2,Y1r\Y2). (3.293) It follows that / is the rank function of a polypseudomatroid.

Q.E.D.

Theorem 3.62 [Chandrasekaran+Kabadi88]: The system of inequalities x{X) - x(Y) < f(X, Y)

((X, Y) € 3E)

(3.294)

is totally dual integral. (Proof) The present theorem follows from Lemma 3.60 and Corollary 3.21. Q.E.D. The concept of polypseudomatroid explained above can be generalized as follows. Let J- be a subset of 3E such that for each (Xi,Yi) € J~ (i = 1,2) the two pairs ((Xx U X2) - ( ^ U y 2 ), (Y1 U y2) - (Xi U X2)) and (X\ n X2, YiC\Y2) belong to !F. Also let / : JF —> R be a function such that for each (XuYi) € T (i = 1,2) / satisfies (3.275). [We call (J^",/) a bisubmodular system on _E.] The class of polypseudomatroids [(bisubmodular polyhedra)] in this generalized sense includes as special cases submodular and supermodular polyhedra, base polyhedra and generalized polymatroids. Further generalization was considered by Qi [Qi88,89]. [For further developments and related results see [Cunningham + GreenKrotki91], [Bouchet + Cunningham95], [Ando + Fuji96], [Ando + Fuji + Nemoto96], [Fuji97], [Ando + Fuji + Naitoh95], [Fuji + Patkar95], [Lovasz97] and [Geelen + Iwata + Murota03]. Similarly as submodular functions on distributive lattices (or ring families), bisubmodular functions on signed ring families are considered in [Ando + Fuji96], where a signed counterpart of Birkhoff's theorem on the poset representations of distributive lattices is also shown (also see [Reiner93]). Moreover, a class of related polyhedra, called polybasic polyhedra, that have edge vectors of support

112

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

size at most two was investigated in [Fuji+ Makino+Takabatake+KashiwabaraO4].] (c) Ternary semimodular polyhedra In the preceding subsection 3.5.b we considered the set 3E of all the ordered pairs (X, Y) of disjoint subsets X and Y of E. 3E is a distributive lattice with lattice operations V and A denned as follows. For any (Xi,Yi) £ 3E (z = 1,2) define (xu Yi) v (x2, y2) = (Xi u x2, Y1 n y 2 ),

(3.295)

(Xh Yi) A (X2, y2) = (Xi n X2, Yx U y 2 ).

(3.296)

Identifying each (X,Y) € 3E with a {0,

r x(X,Y) defined by

f 1 (eeX) x(A-,y)(e) = { -1 (eey) [ 0 (e€£-(XUF)),

(3.297)

we see that operations V and A in (3.295) and (3.296) are ordinary ones in vector lattice HE and that 3E is a finite sublattice of ~RE. Let < be the partial order on 3E associated with the distributive lattice E 3 . We call a pair of (Xi,Yi) € 3E (i = 1,2) comparable if we have (X^Yi) < (X2, Y2) or (X2, Y2) <{XuYi). We also use the partial order < for {0, s under the correspondence (3.297). Let V be a sublattice of the distributive lattice 3E and / : V —> R be a submodular function on P , i.e., for each (Xj, Yi) 6 P ( i = l , 2 )

f{X1,Y1) + f{X2,Y2) > f(X1UX2,Y1nY2) where we write f(X,Y) P(/) by P(/) = {x\x£RE,

+ f(X1nX2,Y1UY2),

instead of f((X,Y)).

(3.298)

Also, define a polyhedron

V(X,Y) € P: x(X) - x(y) < f(X,Y)}.

(3.299)

Without any additional condition the polyhedron P(/) defined by (3.299) may be empty. The following theorem shows that a trivial necessary condition for P(/) to be nonempty is also sufficient. Theorem 3.63 [Puji84e]: P(/) is nonempty if and only if /(0,X) + / ( X , 0 ) > O

(3.300)

3.5. Related Polyhedra

113

for each X C E such that (0, X), (X, 0) € V. (Proof) The "only if" part is trivial. We show the "if" part. Suppose (3.300) holds for each X C E such that (0, X), (X, 0) € V. It follows from the Farkas lemma or the linear programming duality theorem that P ( / ) ^ 0 if (and only if) for all positive CKJ € R (i € / ) and (Xj, Y>) £ V (i £ / ) with a finite index set / such that ^2aiX(Xi,Yi)

=0

(3.301)

Y,<*if(xi,Yi)>0-

(3.302)

iei

we have

Here, note that x(Xi, "Kj) is the coefficient vector of the inequality x{Xi) — x{Y{) < f(Xi,Yi). Since the vectors x(X,Y) ({X,Y) € V) are integral, we can restrict the positive real coefficients ai (i £ I) in (3.301) to positive integers. It follows that P ( / ) ^ 0 if (and only if) for all (Xt,Yt) ef>(iel) with a finite index set / such that

Y/x(Xt,Yt) = O

(3.303)

iei

we have £/(*i.*i)>0,

(3.304)

iei

where possibly (Xi,Yi) = (Xj,Yj) for distinct i,j € / . If t h e family of (Xi,Yi) (i G / ) in (3.303) contains an incomparable pair of (Xi,Yi) a n d (X2,Y2), then put (X1,Yl)^(X1UX2,Y1nY2),

(3.305a)

(X2, Y2) <- (Xi n X2, Yl U y 2 ).

(3.305b)

After the replacement (3.305) relation (3.303) remains valid and the value of the left-hand side of (3.304) does not increase, due to the submodularity of / . Since we get a family of (Xi, Y{) (i € / ) composed of pairwise comparable elements of V after a finite number of such replacements, it suffices to show that we have (3.304) for all (X^Yi) € V (i € I) such that {Xi,Y{) (i € / ) are pairwise comparable and (3.303) holds. Therefore, suppose that (3.303) , m}) and holds for (Xi, Y$ (i € / = {1, 2, (XuYj

X (X2,Y2)

X . . . X (Xm,Ym).

(3.306)

114

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SYSTEMS AND BASE

POLYHEDRA

We see from (3.303) and (3.306) that Xi = 0,

Ym = 0.

(3.307)

Suppose | / | > 2. If Xm — Yi ^ 0, then for any e G Xm — Y\ we have Ylx(XuYi)(e)>0,

(3.308)

which contradicts (3.303). Similarly, Y\ — Xm ^ 0 leads to a contradiction. Hence, we must have (3.309) Yi = Xm. Putting Z1 = Yi{= Xm), we have from (3.307) and (3.309) x ( * i , Yi) + x(Xm, Ym) = X (0, Zx) +X(Zi, 0) = 0.

(3.310)

From (3.310) and assumption (3.300), /(XuYi) + f(Xm, Ym) = /(0, Zx) + f(Z1, 0) > 0.

(3.311)

Put / <— / — { l , m } . For the new index set / (3.303) still holds. Repeat the above argument until | / | = 0 or 1. When | / | = 1, suppose / = {io}Then, from (3.303), = 0, (3.312) X(Xl0,Yl0) Xi0 = 0,

Yi0 = 0.

(3.313)

Hence, from (3.300), / p Q 0 , i y = /(0,0)>O.

(3.314)

It follows from (3.311) and (3.314) that Ytf(Xi,Yi)>0,

(3.315)

i€l

where / = {1, 2,

, m}, the original index set.

Q.E.D.

Corollary 3.64: Define a sublattice VQ off) by Vo = {(X, 0) | (X, 0) G V} U {(0, X) | (0, X) G V).

(3.316)

3.5. Related Polyhedra

115

Let /o be the restriction of f to VQ, and P(/o) be the polyhedron given by (3.299) with V and f replaced by Vo and f0, respectively. Then, P(/) ^ 0 ifandonlyifP(fo) + 0. Corollary 3.64 means that when P(/o) ^ 0, the inequalities in (3.299) with X ^$ and Y ^ 0 do not cut off P(/ o ) too much to give empty P(/). Moreover, it should be noted that if (0, 0) € V and /(0, 0) = 0, then for each (X, Y) G V f(X, Y) = f(X, Y) + /(0, 0) > f(X, 0) + /(0, Y),

(3.317)

so that the inequalities appearing in the right-hand side of (3.299) for (X,Y) <E V with X / 0 and y / 0 are redundant, i.e., P(/) = P(/ o ), where /o is defined in Corollary 3.64. Now, let us consider the following linear programming problem (P) with c(ERE. (P): Maximize

^ c{e)x(e)

(3.318a)

eeE

subject to V(X, Y) € P: x(X) - x(Y) < f(X, Y). (3.318b) In general, the system of linear inequalities (3.318b) is not totally dual integral, and even if/ is integer-valued, P(/) may have non-integral extreme points. For example, consider £ = {1,2},

(3.319a)

P = {({1,2},0),({1},{2})}, /({1,2},0) = 1,

(3.319b)

/({l},{2}) = 0.

(3.319c)

Then P(/) has the only one vertex given by (\1\). The linear programming dual of Problem (P) is described as follows. (P*): Minimize

subject to

^ X(X,Y)f(X,Y) (x,Y)ex>

Y^

(3.320a)

A X Y

A X Y

( ' ) = c(e)

( > ) ~ Y^

(X,Y)ET> eeX

(X,Y)et> e£Y

(e € E), X(X, Y)>0

((X, Y) € V).

(3.320b) (3.320c)

116

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

Though the system of linear inequalities (3.318b) is not totally dual integral, we have the following. Lemma 3.65: Let c 6 HE be an integral vector and suppose that Problem (P*) has an optimal solution. Then there exists a rational optimal solution X*(X,Y) {{X,Y) € V) of(P*) such that £ = {(X, Y) | X*(X, Y) > 0}

(3.321)

consists of pairwise comparable elements ofD. (Proof) By the assumption there exists a rational optimal solution X*(X, Y) ((X, Y) E V) of (P*). For some positive rational d0 each X*(X, Y) {{X, Y) € V) is a nonnegative integral multiple of doIf £ given by (3.321) contains an incomparable pair of elements (Xi, Yi) (i = 1,2), then define a = mm{X*(Xi, Yt) | i = 1, 2} > 0

(3.322)

and change the values of A * ^ , ^ ) (i = 1,2), A*(Xi U X2,Yx D Y2) and A * ( X i n x 2 ) y i u y 2 ) as X*(Xi,Yi) <- A*(X,,y,)-a

(i = 1,2),

(3.323)

A*(Xi U X2, Yl n Y2) <- X*(Xl U X2, Yl n Y2) + a,

(3.324)

\*{Xx n X2, Yx U Y2) <- A*(A"i n X2, Yx U y2) + a.

(3.325)

Then, new A* satisfies (3.320b) and (3.320c) and is an optimal solution of (P*) since the objective function is not made increase by the above replacement (3.323)~(3.325) due to the submodularity of/. Repeat (3.322)~(3.325) so long as £ given by (3.321) contains an incomparable pair. Only a finite number of repetitions of (3.322)~(3.325) are possible since (1) all the generated A*'s are distinct, (2) each \*(X,Y) ((X,Y) G V) is nonnegative integral multiple of d0 > 0 and (3) the sum of all X*(X, Y) ((X, Y) € V) is constant. Q.E.D. The above proof is a direct adaptation of a standard proof technique developed by N. Robertson, L. Lovasz [Lovasz76], J. Edmonds and R. Giles [Edm+Giles77] and A. J. Hoffman [Hoffman74]. Now, consider an additional structure of V and / as follows.

3.5. Related Polyhedra

117

(f) If for (Xi,Yi) ef> (i = l,2) (X1,Y1)y(X2,Y2),

(3.326)

Xx n Y2 / 0,

(3.327)

and X2/0

or Y^iD,

(3.328)

then (Xi - Y2, H), (X2, F2 - Xi) € P and /(Xi, Yi) + /(X 2 , r 2 ) > f(Xi - Y2, Yi) + f(X2, Y2 - Xi). (3.329) Note that (3.326)~(3.328) implies that for some e, e' € E we have x(Xi, yi)(e) = x(^2,y 2 )(e) / 0 and xiX^Y^e!) = -X(X2, Y2)(e') + 0.

Theorem 3.66 [Fuji84e]: Suppose V and f satisfy property (f) described by (3.326)~(3.329). Then system (3.318b) of linear inequalities is totally dual integral. To prove Theorem 3.66 we need some lemmas. The proof of the following lemma is similar to the proof technique developed by Robertson et al. Lemma 3.67: Suppose that c € HE is an integral vector, that Problem (P*) has an optimal solution, and that T> and f: T> —> R satisfy the above additional property (f). Then there exists a rational optimal solution X*(X,Y) ((X,Y) € V) of(P*) such that £ given by (3.321) consists of pairwise comparable elements of V and does not contain any (JQ,Yi) (i = 1,2) which satisfy (3.326)~(3.328). (Proof) By Lemma 3.65, let \*(X,Y) ((X,Y) € V) be a rational optimal solution of (P*) such that £ given by (3.321) consists of pairwise comparable elements of T>. Also let do be a positive rational number such that each X*(X, Y) ((X, Y) G V) is a nonnegative integral multiple of doIff contains pQ,Y~j) (i = 1,2) which satisfy (3.326)~(3.328), then let (Xi, Yi) (i = 1,2) be elements of £ for which (t) (3.326)~(3.328) hold and for each (X3,Y3) € £ such that (X^YJ h (X3,Y3)y(X2,Y2)weh<we X1nY2CE-(X3U

Y3).

(3.330)

118

II. SUBMODULAR SYSTEMS AND BASE

POLYHEDRA

(Note t h a t there always exists such a pair of (Xj, Y$) (i = 1,2).) Define a by a = mm{\*(Xh Yt) \ i = 1, 2} > 0. (3.331) Change the values of X*(Xt, Y*) (i = 1, 2), X*(X1 -Y2,Yi) Xx) as A * ^ , ^ ) ^ <**(*«,^«)-"

and A*(X2, Y~2 -

(i = 1,2),

(3.332)

- Y2, Yi) * - A*(Xi - y 2 , Y{) + a,

(3.333)

A*(X2, Y2 - Xi) <- A*(X2, y 2 - Xi) + a.

(3.334)

\*{X

The new A* satisfies (3.320b) and (3.320c) and is an optimal solution of (P*) because of (3.329). It follows from (3.330) that the new A* gives S in (3.321) which consists of pairwise comparable elements of V. Repeat (3.331)~(3.334) so long as £ in (3.321) contains (X;,Y"j) (i = 1,2) which satisfy the above (J). Then, all the generated A*s are distinct, because each changing of A* by (3.332)~(3.334) increases the total sum of \E - (XL) Y)\X*(X,Y) ( ( I T ) e V) by 2\XX n Y2\a > 0. Also each X*(X,Y) ((X, Y) E T>) is a nonnegative integral multiple of do > 0, and the sum of X*(X,Y) ((X,Y) G V) is constant. Therefore, after a finite number of steps we get a desired optimal solution A* of (P*). Q.E.D. Let A = (a(i,j) \ i £ I, j G J) be a p x q matrix with a row index set / = {1, 2, ,p} and a column index set J = {1, 2, , q}. For an integer t, 7 we say A has the upper consecutive 1 s property (or the lower consecutive t 's property) if a(io,jo) = t for a pair of IQ 6 / and jo € J implies a{i,jo) = t for all i G 7 with i < io (or i > io). Also, we say A has the consecutive t's property

if a(ii,jo)

= a(i2, jo) = i for s o m e

2

G 7 a n d j o £ <7 w i t h

i\ < i2 implies a(i,jo) = t for all i G 7 with ii < i < i2. Denote the ith row and the j t h column of A = (a(i,j) \ i G 7, j G J ) by a(i, and a(-,j), respectively. A rational matrix yl is called totally unimodular if the determinant of every square submatrix of A is equal to 0, 1 or —1. L e m m a 3 . 6 8 : L e tI = { 1 , 2 , (a(i,j) | i G 7, j £ J) is a {0, (i) any pair of rows a(i\,

, p } a n dJ = { 1 , 2 , x surfi that

and a(i2,

,q}. Suppose A =

of A (i\, i2 G 7) is comparable and

3.5. Related Polyhedra

119

(ii) A does not contain, as a submatrix, any of the following 2x2 matrices

(

:

-

!

)

(

-

'

-

'

)

and the ones obtained from these matrices by row and column permutations. Then A is totally unimodular. (Proof) Suppose that the rows of A are indexed so that a(p,

) r< a(p - 1, ) r<

^ a(l,

(3.335)

Also suppose q = p. We show that the possible values of the determinant of the (square) matrix A are 0, 1 and — 1. Since the following argument is valid for any square submatrix of the original matrix A, this implies that A is totally unimodular. Define /(+) = {{ | iel, O^a(v)}, (3.336) / ( - ) = {« \i^I, / ( + , - ) = {1, 2,

a(v)r<0},

(3.337)

,p} - (/(+) U / ( - ) ) ,

(3.338)

where 0 is the zero row vector of dimension | J\(= |/|). From (3.335), A has the upper consecutive l's property and the lower consecutive — l's property. If /(+) U / ( + , - ) = 0, then A is totally unimodular and detA = 0, 1 or —1, since A is a {0,-1} matrix and has the consecutive —l's property [Hoffman+Kruskal56]. Therefore, suppose /(+) U/(+,—) 7^ 0. Let a(-,jo) be a column of A such that a(i,jo) = 1 for all i € I(+) U / ( + , —). (Such a column exists because of (3.336), (3.338) and the upper consecutive l's property of A.) If a(io, Jo) = —1 for some IQ € /(—), then / ( + , —) = 0, since otherwise for an arbitrary i\ € /(+,—) we have a(ii,jo) = 1 and there exists a column a(-,j\) such that a(i\,ji) = a(io,ji) = —1, which contradicts the assumption. When / ( + , —) = 0, A is totally unimodular and detA = 0, 1 or —1, since by multiplying each row a(i, ) (i € /(—)) by — 1, A becomes a matrix which has the consecutive l's property if the rows are appropriately reordered. Therefore, further suppose a(i,jo) = 0 for all ieJ(-). Now, transform the matrix A by fundamental row operations with pivot a(l, jo) = 1 in such a way that the joth column a(-,jo) becomes a unit

120

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA

vector. Let A' be the matrix obtained from the resultant matrix by further removing the first row and the joth column, and let A[ and A'2 be the submatrices of A' composed, respectively, of row vectors corresponding to (/(+) U/(+, —)) — {1} and /(—). Then, since A has the upper consecutive l's property and the lower consecutive — l's property and since by the assumption for any j € J such that a(l,j) = 1 we have a(i,j) = 0 or 1 for all % € /(+) U / ( + , —), both A\ and A'2 have the lower consecutive —l's property. Hence, A' has the consecutive —l's property by appropriately reordering the rows (i.e., by reversing the row ordering of A'2) and is totally unimodular. We thus have detA = ztdetA' = 0, 1, or —1. Q.E.D. We are now ready to prove Theorem 3.66. (Proof of Theorem 3.66) Let c € HE be an integral vector, and suppose Problem (P*) has an optimal solution. By Lemma 3.67 there exists an optimal solution A* of (P*) such that £ given by (3.321) consists of pairwise comparable elements of V and does not contain any (Xi, Yi) (i = 1, 2) which satisfy (3.326)~(3.328). Then it follows from Lemma 3.68 that the set of row vectors x{X, Y) ((X, Y) £ £) forms a totally unimodular matrix, which implies the existence of an integral optimal solution of (P*). Q.E.D. We call the polyhedron P(/) the ternary semimodular polyhedron associated with (£>,/) satisfying property (f). A general framework for proving total dual integrality of systems of linear inequalities related to submodular functions is proposed by A. Schrijver [Schrijver84a,84b], which includes total dual integrality of the EdmondsGiles submodular flow model [Edm + Giles77] and many other models. However, it is not clear whether the total dual integrality of (3.318b) with property (f) of (3.326)~(3.329) follows from Schrijver's framework. For a submodular system (T>',f) and a supermodular system (T>",g") on E, define V C 3E by V = {(X, 0 ) | X € V'} U { ( 0 , Y ) \ y £ V"}

(3.339)

and for each (X, Y) € V

f(X Y) - I f(X,Y)-^_g/f{Y)

f{ X)

-

if Y

= 0'

i { x = t

(3 340) (3.340)

3.5. Related Polyhedra

121

Then, / : P —> R is a submodular function on P, and P(/) denned by (3.299) is expressed as

P(/) = P(/')nP(s")>

(3-341)

where P(/') is the submodular polyhedron associated with the submodular system (P', /') and P(g") is the supermodular polyhedron associated with the supermodular system (V",g"). Theorem 3.63 implies that P(/')nP(#") is nonempty if and only if g"(X) < f'{X) for each X £ V nT>". Moreover, it should be noted that there exists no pair of (Xi,Y{) in the present P which satisfies (3.326)^(3.328). Therefore, if / ' and g" are integer-valued and P(f')nP(g") ^ 0, each face of P(f')nP(g") contains integral points due to Theorem 3.66. This leads to the discrete separation theorem (Theorem 4.12) due to [Frank82b]. It should be noted that a submodular polyhedron, a supermodular polyhedron, a base polyhedron, and the intersection of two base polyhedra are special cases of the intersection of submodular and supermodular polyhedra and that all of these polyhedra are ternary semimodular polyhedra. Hence, in general the greedy algorithm does not work for ternary semimodular polyhedra. Let (V, f) be a submodular system and (V",g") a supermodular system, both on E, such that P(f',g") = {x\xenE,VXe

V: x(X) <

f'(x),

VY € V": x(Y) > g"(Y)}

(3.342)

is a generalized polymatroid, i.e., for each X E T>' and Y € T>" we have X -Y € P',

Y-X

€ P",

(3.343)

/'(X) - g"(Y) > f'(X -Y)g"(Y - X). (3.344) Define V and / : P -^ R by (3.339) and (3.340), respectively. Since /(0, 0) = f(0) = g"(jD) = 0 by definition, we have from (3.344) /(0, X) + f(X, 0) > 2/(0, 0) = 0

(3.345)

for each X £ I?' n 2?". It follows from Theorem 3.63 that generalized polymatroids are always nonempty (cf. Theorems 2.3 and 3.58). Also, the total dual integrality of (3.342) follows from Theorem 3.66.

122

II. SUBMODULAR SYSTEMS AND BASE

POLYHEDRA

Suppose that a distributive lattice T> —> R satisfy the property that for any (Xi,Yi) € P (i = 1,2) satisfying (3.326) and (3.327) we have (X1 - Y2,Yi), (X2,Y2 - Xx) £ t> and (3.329) holds. Then, P ( / ) denned by (3.299) is called a hybrid independence polyhedron by Tomizawa [Tomi81a]. In this case, P ( / ) / 0 if and only if /(0,0) > 0 (if (0,0) € f>), since (3.345) holds for each X such that (0,X), ( X , 0 ) € P . It should be noted that the class of hybrid independence polyhedra includes the class of generalized polymatroids but not the class of intersections of submodular and supermodular polyhedra. Readers should be referred to a nice review [Schrijver84a] for other related polyhedra and systems such as lattice polyhedra ([Hoffman+Schwartz 78], [Hoffman78], [Groflin+Hoffman82]), Frank's kernel system ([Frank79]) and Grishuhin's model ([Grishuhin81]). [Also see a recent excellent book [SchrijverO3].]

3.6. Submodular Systems of Network Type [Tomi+Fuji81] Let J\f = (G = (V,A),c) be a capacitated network with an underlying graph G = (V, A) and a nonnegative upper capacity function c: A — R + , where the lower capacity function is regarded as the zero function. The cut function KC: 2V —> R associated with the network J\f is given by
Y,

KC{U)=

(UCV),

(3.346)

a€A+U

where A+U is the set of arcs leaving U (see Section 2.3). Without loss of generality we assume that G is a simple graph and that each arc a E A is identified with the ordered pair (<9+a, d~a) of its end-vertices. (3.346) can be rewritten as

*c(U)=J2

E

c(u,v)-

ueUv€V-{u}

Y, (c(u,v) + c(v,u))

(UCV),

{u,v}^)

(3.347) where for each arc (u,v) € A we write c((u,v)) as c(u,v), ( 2 ) is the set of all the two-element subsets of U, and we define c(u, v) = 0 for (u, v) ^ A. For any finite set X and any integer i with 0 < i < \X\ we denote by ( i ) the set of all the i-element subsets of X.

3.6. Submodular Systems of Network Type

123

For any non-zero set function / : 2V —> R there exist functions /'*': ( i ) -+ R (0 < i < \V\) such that 1*1

/(*) = £

E

/W(y)

(^V).

(3.348)

By the Mobius inversion formula f^'> (0 < i < |y|) are uniquely determined from / as fW(X)= ]T ( - l ) l x - y l / ( y ) . (3.349) YCX

Let /c be an integer such that 0 < k < \V\, f{k) ^ 0 and / ^ = 0 (A; + 1 < % < \V\). The function / is called a set function of order k (see [Tomi80b]). We see from (3.347) that any cut function is of order 2 if c ^ 0 (also see (3.353) and (3.354) below). A submodular system (2 y , / ) is said to be of network type if / is equal to the cut function KC: 2V —> R associated with a network having a nonnegative capacity function c. Theorem 3.69 [Tomi+Fuji81]: Suppose that ( 2 y , / ) is a submodular system on V. (2V, / ) is of network type if and only if the following (i)~(iii) hold: (i) The order of f is less than or equal to 2. (ii) For the functions / « (0 < i < \V\) in (3.348),

/(°) = 0,

/W > 0,

(3.350)

E / ( 1 ) ( W ) = " E / (2) (^)-

(3-351)

E / ( 1 ) ( W ) < - E /(2)(^)-

(3-352)

(iii) For each X
uex

ue(vj

Unx^V)

(Proof) The "only if" part: If (2V, / ) is of network type and / = KC for a network with a nonnegative capacity function c, then from (3.347) we have

/(°>=0,

/(1)(W)=

Y, v€V-{u}

C

(V),

(3.353)

124

II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA / (2) ({u, v}) = -(c(u, v) + c(v, u)).

(3.354)

Since the capacity function c is nonnegative, (i)~(iii) easily follow from (3.353) and (3.354). The "if" part: Suppose (i)~(iii) hold. Consider the bipartite graph G = iyV+, W~;A) with the left and right vertex sets W+ and W~ and with the arc set A defined by W+ = V,

A={(u,U)

W-=(V\,

| «e£/6

j } .

(3.355)

(3.356)

Let J\f = (G, c) be a network with the underlying graph G and a nonnegative capacity function c such that for each a € A c(a) is sufficiently large, where W+ is the entrance vertex set and W~ the exit vertex set of M. Then, it follows from the assumption that there exists a flow ip: A —> R in J\f such that d
(u£W+(=V)),

d
(3.357)

(3.358)

due to the supply-demand theorem for bipartite networks (Theorem 1.4) ([Gale 57], [Ford + Fulkerson62]). Choose one such flow
((u,w)GA).

(3.360)

Let A/" be the network with the underlying graph G = (V,A) and the nonnegative capacity function c defined by (3.359) and (3.360). From (3.357)~(3.360) we have (3.353) and (3.354). Since the order of / is less than or equal to 2, / coincides with the cut function KC associated with the network M. Q.E.D. We see from this theorem that for a submodular system (2V, / ) on V with the rank function of order at most 2 the problem of discerning whether the submodular system (2V, / ) is of network type or not can be solved by a max-flow computation for the network J\f defined in the above proof, since

3.6. Submodular Systems of Network Type

125

(ii) in Theorem 3.69 can directly be checked and (iii) together with (3.351) is a necessary and sufficient condition for the existence of a feasible flow in the bipartite network A/". Moreover, when (2V, /) is of network type, consider the minimum-cost network-realization problem defined as follows: Find a network M = (G = (V, A), c) with KC = f such that the cost £ 7(a)c(a)

(3.361)

a€A

is as small as possible, where 7(0) is the realization cost per unit capacity for each arc a G i . As is seen from the above proof, this problem is reduced to a Hitchcock transportation problem for the bipartite network J\f defined in the above proof (see [Tomi+Fuji81]).

This Page is Intentionally Left Blank

127

Chapter III.

Neoflows

In this chapter we consider generalizations of classical flow problems of Ford and Fulkerson to flow problems with boundary constraints described by submodular functions. The new flow problems to be treated are the submodular flow problem, the independent flow problem and the polymatroidal flow problem, and they are, in a sense, equivalent. Because of this we call the class of these flow problems and other possible equivalent ones the neoflow problem and each of them a neoflow problem (cf. [Fuji87a]). We give a theory and algorithms for the neoflow problem.

4. The Intersection Problem In this section we consider the problem of finding a maximum common subbase of two submodular systems and some related problems. 4.1. The Intersection Theorem Let (T>i,fi) (i = 1,2) be two submodular systems on E and consider the following problem. P\: Maximize x{E) subject to x € P ( / i ) fl P(/ 2 )-

(4.1a) (4.1b)

Equivalently, we express this problem as follows. Let E' be a copy of E and we regard (X>2,/2) as a submodular system on E'. Also let G = (E,E';A) be the bipartite graph with the left and right vertex sets E and E' and with the arc set A = {(e, e') | e € E}, where e' E E' is a copy of e E E, i.e., A gives a natural bijection between E and its copy E' (see Fig. 4.1). Furthermore, we consider a capacity function c: A ^ R U {+00} such that c(a) = +00 (a E A). Then Problem Pi in (4.1) is equivalent to the following problem Pi': Maximize dip{E) subject to (d
(4.2a) (4.2b) (4.2c)

where ip: A —> R is a flow, dtp is the boundary of ip in Af and (dip)E and (dip)E are, respectively, the restrictions of dip to E and E'. A flow

128

III. NEOFLOWS

Figure 4.1: The intersection problem. (p: A —» R satisfying (4.2b) and (4.2c) is called a feasible flow in A/" = (G = (E,E';A),c,S1,S2), where Sj = (£>i,/j) (i = 1,2). As we will see later, Problem Pi' is a special case of a neoflow problem.

(a) Preliminaries We need some preliminaries to furnish an algorithm for solving the intersection problem described by (4.1) or (4.2). Consider a submodular system (T>, f) on E. In the following we give several lemmas which are obtained by a direct adaptation of the results shown in [Fuji78a] for polymatroids. Lemma 4.1: Suppose x £ P(/), u £ sat(x) and v £ dep(x,u) — {u}. For any a € R such that 0 < a < c(x, u, v) define y = x + a(xu ~ Xv)- Then, y € P(/) and sat(y) = sat(x-). (4.3) (Proof) From the definition of the exchange capacity we have y E P(/)Also, since for any X € V such that X D sat(x-) we have y(X) = x(X) and sat(x) is the unique maximal tight set X such that x(X) = f(X), we have sat(y) = sat(x). Q.E.D.

4.1. The Intersection Theorem

129

Lemma 4.2: Under the same assumption as in Lemma 4.1, c(y, w) = c(x, w)

(w € E — sat(z)).

(4.4)

(Proof) Put X o = sat(z)(= sat(y)). Since x(X0) = f(X0) and y(X0) = f(Xo), we have for any I g P

f(X)-y(X)

= f(X)-y(X)+f(X0)-y(X0) > f(X U Xo) - y(X U Xo) + f{X n Xo) - y(X n Xo) >

/(X U Xo) - y(X U Xo)

=

/(XUXo)-x(XUXo),

(4.5)

and similarly, /(X)-x(X)>/(XUX0)-x(XUX0). Hence the lemma follows from (2.37).

(4.6) Q.E.D.

Lemma 4.3: For any x 6 P ( / ) let u\, u^ and i>2 be three distinct elements of E such that (a = 1,2), (4.7) Ui € sat(x) t>2 € dep(x,tt2),

t>2 ^ dep(x,«i).

(4-8)

For any a € R, such that 0 < a < c(x, it2, ^2) define y = x + a(xu2-Xv2)-

(4-9)

Tien we Lave «i G sat(y) and dep(y, ui) = dep(x, u{).

(4.10)

(Proof) From Lemma 4.1 we have u\ 6 sat(y). Also we have ui (£ dep(x, u\), since otherwise we would have dep(x, U2) C dep(x-, «i) by the minimality of dep(x, ^2) and hence vi € dep(x, u\). Therefore, putting Xo = dep(x, u\), we have y(X0) = x(X0) = /(X o ) and y(X) = x(X) for any X € P with X C Xo. (4.10) follows from the definition of dependence function. Q.E.D. Lemma 4.4: For any x G P ( / ) let u\, u
(4.11)

130

III. NEOFLOWS

Then for the vector y defined by (4.9) for any a € R with 0 < a < c(x, U2, V2), we have (4-12) c(y, ui,vi) = c(x, ui,vi). (Proof) For any z € P(/) and I

o

e P such that z(X0) = f(X0) we have

f(x)-z(x)>f(xnxo)-z(xnxo)

{Xev).

(4.13)

For XQ = dep(x, ui) we have from Lemma 4.3 y(X0) = x(X0) = f(X0)

(4.14)

and since U2, V2 ^ dep(x,iti), we have y(X) = x(X)

(X C Xo, X € P).

(4.15)

Since (4.13) holds for z = x, y, (4.12) follows from (4.14) and (4.15). Q.E.D. L e m m a 4 . 5 : For any x € P ( / ) let Ui, V{ (i = l,2,---,q) elements of E such that Ui € sat(x),

wj € dep(x, ui)

vj £ dep(z, Ui) For a n j ttj (i = 1,2, de&ne a vector y € R

(i = 1,2,

be 2q distinct

, q),

(4.16)

(l
(4.17)

, q) satisfying 0 < ai < c(x, Ui, v{) (i = 1,2, by

, q)

Q

y = x + Y^at(xuz-XvJ-

(4.18)

Tiien, y e P(/),

(4.19)

sat(y) = sat(x),

(4.20)

c(y,w) = c(x,w)

(w
(4-21)

(Proof) Considering the elementary transformations in the order of the pairs (uq,Vq), (uq-i,Vq-i), , (ui,vi), the present lemma can be shown by repeatedly applying Lemmas 4.1~4.4. Q.E.D.

4.1. The Intersection

Theorem

131

It should be noted that we have (4.19)~(4.21) if (4.16) and (4.17) hold for an appropriate numbering of itj's and v^s. Lemma 4.5 will play a very fundamental role in developing algorithms for solving the intersection problem and other related problems.

(b) An algorithm and the intersection theorem We now consider Problem P\ described by (4.2). Given a feasible flow tp in network J\f = (G = (E, E';A), c, Si, S2), define the auxiliary network J\ftp = {Gip = (V, Ay),^) associated with ip as follows. Gv = (V, Av) is a directed graph, called the auxiliary graph associated with ip, with vertex set V and arc set A^ given by V Av

= EUE'u{s+,s-},

(4.22)

= S+UA+UA*UB*UA-US~,

(4.23)

where S+

= {(s+,v)\v(EE-sat+{d+v)}, +

(4.24)

+

+

+

A+

= {(u,v) I vE sat (d
(4.25)

A*

= A,

(4.26)

B*

= {(e',e)

e<EE},

A~

= {(u,v)

u E sat~(d~
5-

= {(v,s-)\veE-sat-(d~tp)}.

(4.27)

(4.29)

Here, d+(p = (dip)E, d~ip = —{di,/i) on E ((£"25/2) on E'). Note that B* is the set of the reorientations of arcs of A. We also define the capacity function c^: A^ - > R U {+00} by ' c+(d+ip,v) c+(d+
(a=(s+,v)ES+), (a=(u,v)eA+), (a G A* U B*), (a= (u,v) G A~), (a=(v,s~) eS~),

(4.30)

III. NEOFLOWS

132

Figure 4.2: An auxiliary network. where c + and c + (c and c ) are, respectively, the saturation capacity and the exchange capacity denned with respect to submodular system (V\, f\) on E ((T>2, fa) on E1). Figure 4.2 shows the auxiliary network A/^. Let us consider an algorithm described as follows. We call a directed path from s+ to s~ of the minimum number of arcs a shortest path from s+ to s~. We assume an oracle for exchange capacities for (2?,, fi) (i = 1,2).

Algorithm for the intersection problem Input: a feasible flow ip in M = (G = (E, E'\ A),c, Su S 2 ). Output: a maximum flow ip in M. Step 1: Construct the auxiliary network J\fv = (G^ = (V, Av),dp). Step 2: If there exists no directed path from s+ to s~ in A/"^, then the algorithm terminates and the current if is a maximum flow in J\f. Otherwise, find a shortest path P from s+ to s~ in Afv and put a <— mm{ccp(a) \ a is an arc in P}, m(Wn\ ^- S

^

^(°) + a

I (p(a) -a

( a = (e' e') e A and a is in P ) '

(4-31)

a qo\

(a = (e, e') e A and (e', e) is in P)>

>

4.1. The Intersection

Theorem

133

Go to Step 1. (End)

Theorem 4.6: The flows ip obtained in Step 2 of the algorithm are feasible flows in M. Moreover, each time we carry out (4.31) and (4.32), the How value of (p increases by a > 0 given by (4.31).

(Proof) In Step 2 we find a shortest path P from s + to s in the auxiliary network Mv. Let (uf, vf), (tt+, v+) be the arcs in A+ lying on P in this order and (uj~, uj~), , (u^, v~) be the arcs in A^ lying on P in this order. By the way of choosing path P, for each i, j such that 1 < i < j' < p (g) we have u+ ^ dep + (a + ^, ut)

( ^ ^ dep-(a-v3, «r)).

Therefore, the present theorem follows from Lemma 4.5.

Figure 4.3: The reachable set U*.

(4.33) Q.E.D.

134

III. NEOFLOWS

Theorem 4.7: If there exists no directed path from s+ to s~ in the auxiliary network M^ in Step 2 of the algorithm, the current flow ip is a maximum flow in M. (Proof) Let U* be the set of the vertices in V which are reachable from s + along directed paths in Mv (see Fig. 4.3). Then for each v E E - U* and u € E' fi U* we have dep+(d+f,v)

C E-U*,

(4.34)

dep-(d-(p,u) C E'nU*.

(4.35)

d+
(4.36) (4.37)

Therefore,

Moreover, from the definition of U* and the auxiliary network J\fv we see EDU* = {e | e' € E'nU*} (C E).

(4.38)

From (4.36)^(4.38), d+ip(E) = d+
(4.39)

On the other hand, for any feasible flow

= d+0(E - U) + d-
+ f2{E'nU).

(4.40)

It follows from (4.39) and (4.40) that the current flow ip is a maximum flow in M. Q.E.D. Since there exists a maximum flow in M (this fact will also algorithmically be proven later), Theorems 4.6 and 4.7 together with the proof of Theorem 4.7 show the following theorem. Note that when f\ and f2 are integer-valued, starting with an integral feasible flow in J\f, the algorithm given above terminates after a finite number of steps and then gives an integral maximum flow.

4.1. The Intersection Theorem

135

Theorem 4.8: For Problem Pi' described by (4.2) we have max{<9 + ip(E) | ip: a feasible flow in jV} = m i n { / i ( J 0 + f2(Ef

-X')\X£

P i , E' - X' € P 2 } ,

(4.41)

where X' = {x' \ x £ X} C E'. Moreover, if f\ and / 2 are integer-valued, there exists an integral maximum Bow in J\f. Rewriting Theorem 4.8 for Problem Pi, we also have Theorem 4.9 (The Intersection Theorem): For Problem Pi described by (4-1), max{x(E) | x € P(/i) D P(/ 2 )} = min{/i(X) + f2(E -X)\X EVU E-X

€ V2}.

(4.42)

Moreover, if / i and /2 are integer-valued, the maximum on the left-hand side of (4.42) is attained by an integral vector x £ P ( / i ) n P(f2)Theorem 4.9 is a generalization of the intersection theorem for polymatroids due to Edmonds [Edm70]. The algorithm given in this section is a direct adaptation of the one given in [Puji78a]. Denote by C the set of all the minimizers of the right-hand side of (4.42). £ is a sublattice of T>\ n T>2. Then, we have the following theorem. Recall that for a submodular system (Z>, / ) and X,Y GV with I c F / } denotes the rank function of the minor (P, / ) Y/X. Theorem 4.10 (cf. [Iri79], [Nakamura+Iri81]): Let C: S~ = 5i C S2 C

C S k = S+

(4.43)

be any (maximal) chain of C and define So = 0,

Sk+l = E.

Then, the following two statements are equivalent: (i) x is a maximum common subbase of iT>\, ji) [i = 1, 2).

(4.44)

136

III. NEOFLOWS

(ii) For each i = 1, 2, , k + 1, x ^ - ^ - i is a maximum common subbase of (Pi, / i ) Si/Si-x and (V2, f2) (E - Sl-1)/(E - Sl), where for each i = 1, 2, , k x *~ i-1 is a base of (T>\, f\) S-i/S%-\ and for each i = 2, 3, , k + 1 s s *- s *-i is a base of (V2, / 2 ) (E - St-1)/(E - St). (We disregard the case of i = 1 (or i = k + 1) if Si = 0 (or Sk = E).) (Proof) (i) =^> (ii): Suppose (i). From Theorem 4.9 we have x(Si) + x(E-Sl)

= x(E) = f1(Sl)

Since x(5j) < fi(Si) must have x(50 = /i(5i),

+ h(E-Si)

(i = l,2,---,k).

and x{E - Si) < f2{E - Si) for i = 1, 2,

x(E - Si) = f2(E - Si)

(i = l,2,---,k).

(4.45)

, k, we

(4.46)

Since x e P ( / i ) n P ( / 2 ) , it follows from (4.43) and (4.46) that x5*"5*"1 (i = 1,2, ) are bases of {T>iJi)-Si/Si-i and that a;^"^- 1 (i = 2, 3, , k + 1) are bases of (T>2, (E — Si-i)/{E — Si), where we disregard the case of i = 1 (or i = k + 1) if Si = 0 (or Sk = E). Hence, (ii) holds. (ii) =^ (i): Suppose (ii). Then we have x € P ( / i ) H P(/2)- It follows from (ii) that k

x(E) = x(E-Sk) + Y/z(Si-Sl-1)

+ x(Sl)

k

= h(E- Sk) + E(/i( s O " h(Si-i)) + fi(Si) i=2

= fi{Sk) + h{E-Sk).

(4.47)

From (4.47) and Theorem 4.9 x is a maximum common subbase of (Pj, /j) (i = l,2). Q.E.D. We can also show that the set of the minors of (Pi, / i ) and (P2, / 2 ) appearing in (ii) of Theorem 4.10 does not depend on the choice of a maximal chain C of C ([Nakamura+Iri81], [Tomi+Fuji82]) (see Theorem 7.17). Theorem 4.10 is a generalization of a structure theorem for bipartite graphs [Dulmage+Mendelsohn59]. (c) A refinement of the algorithm

4.1. The Intersection Theorem

137

When / i and f^ are not integer-valued, the algorithm shown in Section 4.1.b may not terminate after finitely many steps. We shall show an algorithm due to Schonsleben [Schonsleben80] and Lawler and Martel [Lawler+Martel 82a] which modifies Step 2 of the algorithm and terminates after repeating Step 2 0 ( | £ | 3 ) times. Let vr: V(= E U E') —> {1, 2, , 2n} be a one-to-one mapping which denotes a numbering of the vertices in V, where n = \E\. We also define 7r(s+) = 0 and vr(s~) = In + 1. We represent any directed path P from s+ to s~ in Nip by the sequence (s + , v\, i>2, , vp, s~) of vertices in P and denote by vr(P) the sequence (TY(VI),TY(V2). ,ir(vp)) of the numbering indices of the intermediate vertices. For any directed paths Pi and P2

from s+ to s~ in J\fv we say Pi is lexicographically smaller than P% if 7r(Pi) is lexicographically smaller than vr(P2). We call the lexicographically smallest one in the set of all the shortest paths from s+ to s~ in Nv the lexicographically shortest path from s + to s~ in A/^> (with respect to 7r). Now, we modify the part of finding a shortest path P in Step 2 of the algorithm as follows. (*) If there exists a directed path from s+ to s~ in A/^, find the lexicographically shortest path P from s+ to s~ in A/^. Theorem 4.11 ([Schonsleben80], [Lawler+Martel82a] for polymatroids): If we modify Step 2 of the algorithm given in Section 4.1.b for Problem P\ as above, the algorithm Ends a maximum Row in M after repeating Step 2 O(|£| 3 ) times. (Proof) Denote by Wk Q V U {s + ,s~} the set of the vertices which are reachable by a directed path from s+ having k arcs but not reachable by a directed path from s+ having less than k arcs. Suppose WQ = {s+} and s~ G Wp. Let P be the lexicographically shortest path in J\fv. Also suppose that the arcs of A^ lying on P appear in order as (u+,v+),

(u+,v+),

-..,

{u+,v+)

(4.48)

from s+ to s~. Put x = d+tp and define y = x + a(Xv+-Xu+),

(4-49)

where a is defined by (4.31) for the current ip. Consider vertices w, z € E such that w G dep+(y,z). (4.50) w ^ dep + (x-,z),

III.

138

NEOFLOWS

Then we must have uf € dep + (x,z),

vf <£ dep + (x,z),

w € dep + (x,vf).

(4.51) (4.52)

(See Fig. 4.4.) Here we may have vf = w or z = uf. dep+(x,u+)

dep+(x,z)

Figure 4.4: Exchangeable pairs. Now, suppose uf e Wit,

^

e Wifc+i

(4.53)

for some k (1 < k < 2n). Then we have z € W^ for some positive integer k\ with fci < k + 1. (1) If vertex w is not reachable from s+ in A/^, then adding arc (w, z) to Mif does not change the set of shortest paths from s + to s~ in A^,. (2) Suppose w E Wk2 for some k2 (1 < k2 < 2n). (Note that fc2 > A;.) (2-1) If &2 > A; or fci < A; + 1, then adding arc (w, z) to A/^ does not change the set of shortest paths from s+ to s~ in M^. (2-2) If k2 = k, k\ = k + 1 and there exists no shortest paths from s+ to s~ which include vertex z, then adding arc (w, z) to M^ does not change the set of shortest paths from s+ to s " in A/^.

4.1. The Intersection Theorem

139

(2-3) If &2 = k, k\ = k + 1 and there exists a shortest path from s+ to s~ in M

(4.54)

since P is the lexicographically shortest path. Because of Lemma 4.5 we can repeatedly apply the above argument to arcs (v+,v+) (i = 2,---,/)in(4.48). We can also apply the argument for A+ to A~ mutatis mutandis. For each vertex w € V r U{s + } define P
min{vr(f) | v £ V, (w,v) lies on a shortest path from s+ to s~ in W^}.

(4.55)

Also denote by ip* the feasible flow obtained from ip by transformation (4.32), and let P* be the lexicographically shortest path in Mv*. Since at least one arc lying on P is missing in J\fv*, it follows from the above argument that (i) (the length of P*) > (the length of P) + l or (ii) (the length of P*) = (the length of P) and P

for some v G V U {s+}.

Case (ii) occurs consecutively O(|y| 2 ) times and hence Step 2 is executed O(|y| 3 ) (=O(|£| 3 )) times. Q.E.D. The above proof technique is due to R. E. Bixby (cf. [Cunningham84]). It should be noted that Theorem 4.11 together with Theorem 4.7 shows the existence of a maximum flow in M for any totally ordered additive group R and that the modified algorithm finds a maximum flow in N by changing feasible flows OdE1]3) times. The above algorithm corresponds to the Edmonds-Karp algorithm for classical maximum flows [Edm + Karp72]. A complexity improvement over the above algorithm is shown in [Tardos + Tovey + Trick86] by generalizing the idea of layered networks due to E. A. Dinits [Dinits70]. [Currently the fastest algorithm for the intersection problem is given by Fujishige and Zhang [Fuji+Zhang92] by adapting the push-relabel algorithm of Goldberg and Tarjan [Goldberg+Tarjan88] for maximum flows.]

140

III. NEOFLOWS

4.2. Discrete Separation Theorem A. Frank showed the following theorem. We prove this theorem by the use of the intersection theorem (Theorem 4.9). (Note that we have already shown this theorem in Section 3.5.c.) Theorem 4.12 (Discrete Separation Theorem) [Frank82b]: Let ( P i , / ) be a submodular system on E and (V2,g) be a supermodular system on E. Then, g
(4.57)

Therefore, there exists a vector x in P ( / ) n B(g) (= P ( / ) n B(^ # )). This vector x satisfies g < x < f. Moreover, if / and g are integer-valued, then from (4.57) and the intersection theorem there exists an integral x € P(/) fi B(g), which satisfies g <x < / . Q.E.D. We can also show the intersection theorem (Theorem 4.9) by using the discrete separation theorem (Theorem 4.12) as follows. Let (T>i, fi) {i = 1,2) be submodular systems on E. Define k = min{/i(X) + f2(E - X) \ X € Pi, E - X e V2}.

(4.58)

Also define / 2 : P 2 - R by f

j h{X)

(leP 2 -{£}),

(459)

Note that f2: V2 —> R is a submodular function since k < f2(E); in fact, (Z>2, h) is the (h(E) - ^-truncation of (P 2 , / 2 ) . Then we have / 2 # (0) =

4.2. Discrete Separation Theorem

141

0 = /i(0) and for each X € Vx n V~2 ~ {0}

/2#P0

= f2(E) - f2(E - X) =

k-f2{E-X)

< fi(X).

(4.60)

It thus follows from the discrete separation theorem that there exists a vector x € R^ such that / 2 # < x < / i . Since P(/i) n P(/2 # ) 7^ 0, we have P(/i)nB(/ 2 ) (= P(/i) nB(/ 2 #)) 0. Consequently, max{x(E) | x € P ( / i ) n P ( / 2 ) } >max{x(£) | x € P(/i) D P(/ 2 )} =k = minihiX) + f2(E-X)\X£Vl,

E-Xe

> max{x(E) I x € P(/i) D P(/ 2 )},

Z>2}

(4.61)

where the last inequality follows from the fact (the weak duality) that x(E) < h(X) + f2(E - X) for any x € P(/i) n P ( / 2 ) and any X € Vx with E - X € V2. This proves (4.42). Moreover, the integrality part of the intersection theorem follows from the integrality part of the discrete separation theorem and the above argument. We have thus shown the equivalence between the intersection theorem and the discrete separation theorem. Using the discrete separation theorem, we show (3.32) and (3.33) in Section 3.1.c of Chapter II. For two submodular systems (T>i, ft) (i = 1, 2) on E let / be the submodular function on T>\ H P 2 defined by

f(x) = h{x) + f2{x)

{x e Pi n v2).

(4.62)

We write / as /i + f2. We can easily see that the relation, P(/i + f2) 5 P(/i)+P(/ 2 ), holds. Conversely, for any x € P(/i + f2) we have

VX € Z>i n Z>2: x(X) < h(X) + f2(X),

(4.63)

which is rewritten as VX € V1 n Z>2: x(X) - / 2 p 0 < /i(X).

(4.64)

142

III. NEOFLOWS

It follows from (4.64) and the discrete separation theorem that there exists a vector y G R £ such that VXEV2:

x(X)-f2(X)
VXePi: y{X)
(4.65) (4.66)

Denning z = x—y, we have from (4.65) z E P(/2) a n d from (4.66) y E P(/i). We thus have x = y + z£ P(/i) + P(/ 2 ). Therefore, P(/i + f2) C P(/i) + P(/ 2 ). This completes the proof of the fact that P(/i + / 2 ) = P ( / i ) + P ( / 2 ) . Since (/! + f2)(E) = h{E) + f2(E), we also have B(/i + / 2 ) = B(/i) + B(/ 2 ). 4.3. The Common Base Problem Let (T>i, fi) (i = 1, 2) be two submodular systems on E. The common base problem for submodular systems (T>i, fi) (i = 1,2) is to discern whether there is a common base x € B(/i) fl B(/ 2 ) and, if any, to find one such common base. Clearly, that fi{E) = f2(E) is necessary for the existence of a common base. Theorem 4.13: Let (T>i, fi) (i = 1,2) be submodular systems on E with fi(E) = f^{E). Then there exists a common base x E B(/i) fl B(/ 2 ) if and only if j2* < fi, i.e.,

yXEV n W2: f2#{X) < h{X).

(4.67)

Moreover, if fi (i = 1, 2) are integer-valued and a common base exists, then there exists an integral common base. (Proof) If there exists a common base x € B(/i)nB(/ 2 ), then since B(/i)n B(/ 2 ) = P(/i) n P(/ 2 #), we have / 2 # < x < / i . Conversely, if / 2 # < fu then there exists a vector x € R £ such that / 2 * < x < / i , due to the discrete separation theorem (Theorem 4.12). Hence x E P(/i) H P(/ 2 *) = B(/i)nB(/2#) = B(/1)nB(/2). Moreover, the integrality part also follows from the integrality part of the discrete separation theorem. Q.E.D. Note that, in Theorem 4.13, / 2 # < f\ if and only if / i # < / 2 , since ft(E) = f2(E). Also note that "f2* < / i " is equivalent to:

4.3. The Common Base Problem

143

\/X € Z>i n V2: h{X) + h(E -X)> f2(E) (= ME)).

(4.68)

From the intersection theorem, (4.68) means that there exists a vector x € P(/i)flP(/2) such that x(E) > f2(E), and such a vector x is a common base since fi(E) = f2(E). In this way Theorem 4.13 can also be shown by the intersection theorem. The common base problem can be solved by rinding a maximum common subbase x € P(/i) ft P(/2) through the algorithm shown in Section 4.1. We shall also give an algorithm for the common base problem which deals only with bases in B(/i) and B(/2). Given bases b\ € B(/i) and b2 € B(/ 2 ), we denote /3 = (61,62) and define the auxiliary network Np = (Gp = (V,Ap),Tt,Tg,cp) as follows. G/3 is the underlying graph with the vertex set V = E and the arc set Ap defined by A?

= ApUA},

(4.69)

A}j = {(u,v) 2

A p = {(u,v)

u,v£ V, u£ dep1(6i,-y) - {v}},

(4.70)

u,v (EV, v <Edep2(b2,u) - {u}},

(4.71)

where depx and dep2 are the dependence functions associated with (Pi, /1) and (T>2,f2), respectively, cp: Ap —> R is the capacity function defined by

C {a)

e

=

f £1(61,w,«) \~c2(b2,u,v)

{a = (u,v)eA},), {a = (u,v)eA%.

( 4 J 2 )

Here, ci and c2 are the exchange capacities associated with ( P i , / i ) and (P2, f2), respectively. Also, Tt and T J are subsets of V defined by T£

=

{v\v<=V, h(v)>b2(v)},

(4.73)

Tp

=

{v\ve

(4.74)

V, bx(y) < b2(v)}.

Tt is the set of entrances and T^ the set of exits in J\fp. Now, an algorithm for finding a common base of B(/i) and B(/ 2 ) is given as follows. An algorithm for finding a common base

144

III. NEOFLOWS

Input: Submodular systems (Vu f{) (i = 1,2) on E with fi(E) = f2(E) and initial bases bi of (T>i, fi) (i = 1, 2). Output: A common base b\ (= 62) if any exists. Step 1: While 61/62? do the following (a)~(c): as(a) Construct the auxiliary network J\fp = (Gp = (V, Ap),Tt,T7,cp) sociated with P = (pi, 62). If there is no directed path from Tt to T J , then stop (there is no common base in B(/i) and B(/2)). (b) Let P be a directed path from Tt to T7 in Gp having the smallest number of arcs and put a <— min{min{c/g(a) | a is an arc on P}, bl(d+P)

- b2(d+P), h(d-p) -

h(d-p)}.

(d+P is the initial vertex of P and d~P the terminal vertex of P.) (c) For each arc a on P, if a = (u, v) 6 A\, then put b\(u) <— b\(u) — a,

bi(v) <— b\(v) + a,

if a = (it, v) G A\, then put 6 2 (tt) <— b2(u) + a,

b2(v) <— b2(v) - a.

Step 2: The current 61 is a common base of B(/i) and B(/2) and the algorithm terminates. (End) Because of the way of choosing a directed path P in (b) of Step 1, the vectors bi and 62 remain bases in B(/i) and B(/2), respectively, due to Lemma 4.5. If the algorithm terminates at (a) of Step 1, then let U be the set of the vertices in Gp which are reachable by directed paths from Tt. It follows from the definition of the auxiliary network Mp that 61 (C7)

=

f1(E)-f1(E-U),

b2(U)

= f2(U).

(4.75) (4.76)

Since bi(U) > b2(U) by the assumption and the definition of U, we have from (4.75) and (4.76)

fi(E)(= f2(E)) > f\(E -U) + f2(U).

(4.77)

5.1. NeoBows

145

From (4.77) (and Theorem 4.9) we see that there exists no common base inB(/i)andB(/2). When /i and fa are integer-valued and initial bases b\ and 62 are integral, bases b\ and 62 obtained during the execution of the above algorithm are integral and hence the algorithm terminates after repeating (a)~(c) of

Step 1 at most h(T£) - b2(T^) times. For general rank functions f\ and fi we adopt the lexicographic ordering technique described in Section 4.1.c ([SchonslebenSO], [Lawler + Martel82a]). When finding a shortest path from Tt to T7 by the breadth-first search, for each u 6 V search arc [u, v) in A\ (or AV) earlier than arc (n, v') in AXp (or Afy if vr(-u) < vr(t/), for a fixed numbering vr: V —> {1,2, , |V|} of V. By this modification the algorithm terminates after repeating Cycle (a)~(c) of Step 1 O(|E\3) times.

5. Neoflows In this section we consider the submodular flow problem, the independent flow problem and the polymatroidal flow problem, which we call neoflow problems. We discuss the equivalence among these neoflow problems and give algorithms for solving them.

5.1. Neoflows We first give the definitions of the submodular flow problem, the independent flow problem and the polymatroidal flow problem.

(a) Submodular flows Let G = (V, A) be a graph with a vertex set V and an arc set A. Also let c: A ^ R U {+00} be an upper capacity function and c: i ^ R U {—00} be a lower capacity function. A function 7: A —> R is a cost function. Let J- C 6 V be a crossing family with 0, V 6 J- and / : J- — R, be a crossingsubmodular function on the crossing family T with /(0) = f(V) = 0. (See Section 2.3 for the definition of crossing-submodular function on a crossing family.) Denote this network by Ms = (G = (V. A),c, c, 7, (T, / ) ) .

146

III. NEOFLOWS

The submodular flow problem considered by Edmonds and Giles [Edm + Giles77] is described as follows. Ps: Minimize ^ J ^f(a)ip(a)

(5.1a)

a,€A

subject to c(a) <
(5.1b) (5.1c)

Here, dip is the boundary of ip with respect to G (see (2.23)) and B ( / ) is the base polyhedron associated with / (see Theorem 2.5), where we assume B ( / ) / 0

\

A feasible ip: A —> R satisfying (5.1b) and (5.1c) is called a submodular flow in A/5 and an optimal solution of the submodular flow problem P$ is called an optimal submodular flow in Ms- (The term "submodular flow" was introduced by Zimmermann [Zimmermann82].)

(b) Independent flows Let G = (V, A; S+, S~) be a graph with a vertex set V, an arc set A, a set S+ of entrances and a set S~ of exits such that S+, S~ C V and S+ n »S~ = 0. Also let c: ^4 —> R U {+00} be an upper capacity function, c: A —> R U {—c»} be a /ou;er capacity function, and 7: A ^ R be a cost function. Moreover, let (V+,f+) be a submodular system on 5 + and (T>~, f~) be a submodular system on S~. Denote this network by A// =

(G={V,A;S+,S-),c,c,'y,(V+,f+),(V-,f-)). The independent flow problem [Fuji78a] is given as follows (also see [McDiarmid75]). For a given v* £ R, Pj: Minimize ^ ^y(a)ip(a)

(5.2a)

a€A

subject to c(a) < tp(a) < c(a) (a £ A),

dip(v) = 0 (v e V- (5+U5-)), (^) S + GP(/+),

s

s

(5.2b)

(5.2c) (5.2d)

r

-W eP(/1,

(5.2e)

9^(5+) = u*.

(5.2f)

Here, (dip) (or {d(p) ) is the restriction of 9^: V —> R to 5 + (or 1 + S ^) and P ( / ) and P ( / ~ ) are, respectively, the submodular polyhedra

5.1. NeoBows

147

associated with (T>+,f+) and (T>~,f~). (The original independent flow problem considered in [Fuji78a] is described in terms of polymatroids and is slightly generalized here to submodular systems.) It should also be noted that if the system of (5.2b)~(5.2f) is feasible, we have min{/+(5' + ), f~(S~)} > v* and that letting (V+, / + ) and + + (V-J-) be, respectively, the (f (S ) - u*)-truncation of ) and the (f~(S~) - v*)-truncation of (£>", / " ) , we can replace (5.2d)~(5.2f) by {d
(5.2g)

-(d
(5.2h)

+

Here, note that dip(S ) = -d(p(S~) due to (5.2c). A feasible flow ip: A —> R satisfying (5.2b)~(5.2f) is called an independent flow of value v* in Mi and an optimal solution of the independent flow problem Pi is called an optimal independent flow of value v* in A//. (c) Polymatroidal flows Let G = (V, A) be a graph with a vertex set V and an arc set A. For each vertex v G V we consider distributive lattices V+ C 2s v and T>~ C 2s v and submodular functions / + : T>^~ —> R and f~: T>~ —> R. Here, we assume 0 € £>+, ® £ T>~ but not necessarily 5+w G X>+, 5"u G P ^ . Also let c: T 4 ^ R U { —c»}bea lower capacity function and 7: A —> R be a cost function. Denote this network by Mp = (G = (V, A), c, 7, (P+, /+)(« € F),(p-,/-)(uGy)). The polymatroidal flow problem, [Hassin.78,82], [Lawler + Martel82a,82b] is given as follows. Pp: Minimize ^

7(a)y?(a)

(5.3a)

subject to c(a) < y(a) (a G A), 0
(we F ) ,

s+v

eP(f+)

^ £ P ( / , 1

(5.3b) (5.3c)

(veV),

(5.3d)

(vGV),

(5.3e)

where 9y? is the boundary of 93 with respect to G, ips v (or ips v) is the restriction of ip: A —> R to <5+u (or 5~ti), and P(/+) = {x I x G R 5 + v , VX G V+: x(X) < f+(X)},

(5.4)

148

III. NEOFLOWS P ( / - ) = {x I x G R ^ , VX e V-:x{X) < f~(X)}.

(5.5)

(The problem is originally denned in terms of polymatroids and is slightly generalized here.) A feasible flow ip: A —> R satisfying (5.3b)~(5.3e) is called a polymatroidal flow in Mp and an optimal solution of Problem Pp is called an optimal polymatroidal flow in Mp. 5.2. The Equivalence of the Neoflow Problems We show the equivalence of the submodular flow problem, the independent flow problem and the polymatroidal flow problem. Here, the equivalence is with respect to the capability of modeling flow problems. Different models may require different oracles for algorithms but we will not go into this matter here. (a) From submodular flows to independent flows Consider the submodular flow problem Pg defined by (5.1). From Theorem 2.5 and the definition of the submodular flow problem we can assume that / appearing in (5.1) is a submodular function on a distributive lattice V
S~ = {s~}.

(5.6)

Then the submodular flow problem Ps is rewritten as Minimize ^

j(a)ip(a)

(5.7a)

aeA

subject to c(a) <
(^)

S+

GB(/),

-(dtpf- e {0}.

(5.7b)

(5.7c) (5.7d)

This is an independent flow problem with the underlying graph G' = (V U {s~},A; S+,S~). (See (5.2b), (5.2c), (5.2g) and (5.2h), where constraint (5.2c) is void.) It is interesting to see that the submodular flow problem looks like a very special case of the independent flow problem (see Fig. 5.1).

5.2. The Equivalence of the NeoBow Problems

149

Figure 5.1: From submodular flows to independent flows.

(b) From independent flows to polymatroidal flows Consider the independent flow problem Pj described by (5.2). Without loss of generality we assume that there is no arc entering S+ or leaving S~ and that f+(S+) = f~(S~) = v*. Let s+ and s~ be new vertices not in G = (V, A;S+,S~). Construct a graph G = (V, A) with vertex set V and arc set A given by V A

= VU{s+,s-},

(5.8) 0

+

= A U A U A~ U A , +

A

+

= {(s ,v)

(5.9) +

veS },

A- = {(v,s-) veS~}, A° = {(s-,s+)}.

(5.10) (5.11) (5.12)

Also define a lower capacity function c: A —> R U {—ex)} and a cost function 7: A -> R by ( c(a) c(a) = < v* [ -00 ^

= (0

(aeA), (a = (s-,s+)), (a € A+ l)A~), \aeA+UA-uA°).

(5.13)

(5 14)

-

150

III. NEOFLOWS

Moreover, for each vertex v € V define polyhedra P+ C R*5 v and P~ C R5~» by f P(/+) (v = s+), (ve{s-}US-), P+ = I (-oo,+oo) ( {x\x(E H5+v, Va € 5+v: x{a) < c{a)} (v € V - S~), (5.15) P(/") p-

=

(« = 8~),

( (-00, +00)

(tje{s + }uS+),

( {x I xe~R5~v, Va€ S-v:x(a)
(v(EV-S+),

(5.16)

where P ( / + ) and P(/~) should, respectively, be regarded as polyhedra in HA and HA under the natural correspondence between S+ and A+ and between S~ and A~. Now, the independent flow problem Pj is rewritten as Minimize ^ ;y{a)Lp{a) aeA

(5.17a)

subject to c(a) <
(5.17b)

dip(v) = 0 (v € F ) , +

^ " € P+

(5.17c)

(wet),

(5.17d)

y? "" € P - (v € F).

(5.17e)

5

We can easily see that for each v € V there exist distributive lattices V+ C 2S+V and P,; C 2S~V and submodular functions / + : P + -^ R and / " : P " ^ R such that 0 € V+ D P " , /+(0) = /"(0) = 0 and P+ = P(/+),

P-=P(/-).

(5.18)

Therefore, the independent flow problem is reduced to a polymatroidal flow problem (see Fig. 5.2). (c) From polymatroidal flows to submodular flows Consider the polymatroidal flow problem Pp defined by (5.3). From the underlying graph G = (V, A) we construct a graph G' = (W, A) with the same arc set, where the vertex set W is given by W = {w+ \a£ A}U{w~

I a € A}

(5.19)

5.2. The Equivalence of the NeoBow Problems

151

Figure 5.2: From independent flows to polymatroidal flows. and we define d+a = w£ and d~a = w~ for each a £ A in G' (see Fig. 5.3). Each arc of G' forms a connected component of G'. Define an upper capacity function cf: A —> RU {+00} and a lower capacity function c': A —> RU{-oo} by c'(a) = +00 (a € A),

d{a) = c(a) (a € A).

(5.20)

Also define W+ = {wt\ae

8+v},

(5.21)

W~ = {w~ I a e 5~v},

(5.22)

+

where 8 and 5~ are with respect to G. Under the natural correspondences between W^~ and 8+v and between W~ and 8~v for each s e V w e regard P(/+) and P(/j~) as polyhedra in R^ 1 and RWu , respectively. Define for each v E V Bv = {y | ye

R " ^ . - , yw^ € P(/+), - y ^ 7 € P(/"), y(W+) + y(W-) = 0} (5.23)

and let B C R w be the direct sum of Bv

B=®

(veV):

Bv .

vev Then, the polymatroidal flow problem is rewritten as

(5.24)

III. NEOFLOWS

152

Figure 5.3: From polymatroidal flows to submodular flows. Minimize ^ -y(a)(p(a)

(5.25a)

aeA

subject to c'(a) <
(a<EA),

(5.25b) (5.25c)

Note that y 6 Bv if and only if VX6P+:

y(X)
(5.26)

VXeP-:

-y(X) < / - ( * ) ,

(5-27)

y(W+) + y(W-) = 0, S+V

(5.28)

5 1

and V~ C 2' " '. The system of (5.26)^(5.28) where we regard V+ C 2 is equivalent to that of (5.26), (5.28) and VX € V-: y(W+) + y(W~ - X) < f~(X).

(5.29)

For each v € V let J-v be the crossing family defined by TV=V+UV^U {W+ UW~},

(5.30)

5.3. Feasibility for Submodular Flows

153

where Vy = {W+ U (W~ - X) \ X E V~}, and let /„: ^ - » R b e defined by

r f+(x) fv{X) = \ f-(W~-X)

(x<=v+), (XeVv),

(5.31)

(X = W+UW~).

[0

Then, fv is a crossing-submodular function on the crossing family J-v and we have Bv = B(/y), a base polyhedron. Therefore, it follows from (5.24) and (5.25) that the polymatroidal flow problem Pp is reduced to a submodular flow problem. It is interesting to see again that the polymatroidal flow problem looks like a very special case of the submodular flow problem.

5.3. Feasibility for Submodular Flows In the previous section we have shown the equivalence of the submodular flow problem, the independent flow problem and the polymatroidal flow problem. Since the submodular flow problem is simple to describe, let us consider as a neoflow problem the submodular flow problem Ps'- Minimize 2_^ rYia)ip{a)

(5.1a)

a€A

subject to c(a) < ip(a) < c(a) (a G A), d

(5.1b) (5.1c)

Due to Theorem 2.5 we assume for simplicity that / is a submodular function on a distributive lattice V C 2V with 0, V € V and /(0) = f(V) = 0. Recall (2.65), where we have shown <9$ = {dip | cp: A -> R, Va e A: c(a) <
B(/Cc,c).

(5.32)

Here, K^C- %V —> RU{+oo} is the cut function associated with the network M = (G = (V,A),c,c) and d& is the base polyhedron associated with the cut function KC^- Hence, there exists a feasible flow ip for the submodular flow problem if and only if 3$nB(/)(=B(^)nB(/))/0, i.e., there exists a common base in B(KCJC) and B ( / ) .

(5.33)

154

III. NEOFLOWS From Theorem 4.13 we have

Theorem 5.1 [Frank84]: There exists a feasible flow for the submodular How problem P$ satisfying (5.1b) and (5.1c) if and only if VX G V:

{K^)*{X)

< f(X)

(5.34)

or VX G V: c(A~X) - c(A+X) + f(X) > 0, +

(5.35)

+

where for each X C V A X = {a \ a G A, d a G X, d~a <E V - X} and A~X = {a | a G A, r « £ X, d+a G V - X}. Moreover, if~c, c and f are integer-valued and P,s is feasible, there exists an integral feasible Bow. (Proof) Immediate from Theorem 4.13.

Q.E.D.

A feasible flow for the submodular flow problem can be obtained by the use of the algorithm shown in Section 4.3. Frank [Frank84] showed feasibility theorems for the cases where / is an intersecting-submodular function and where / is a crossing-submodular function. We can give Frank's result by combining Theorems 5.1 and 2.6. That is, Corollary 5.2 [Frank84]: (i) When f is an intersecting-submodular function on an intersecting family T such that 0, V € T and /(0) = f(V) = 0, the submodular flow problem P$ described by (5.1) has a feasible flow if and only if we have

{^*{X)
(5.36)

iei

for each X C F and disjoint Xi G J- (i G / ) such that X = Ujg/ -^i(ii) When f is a crossing-submodular function on a crossing family J- such that 0, V G T and /(0) = f(V) = 0, the submodular flow problem Ps has a feasible flow if and only if we have

(^cf(X)
(5-37)

iei jeJi

for each X C V, codisjoint Xi Q V (i E I) and disjoint Xij (j G Jj) (for each i G / ) such that X = f]ieI Xi and Xi = [jjeJi Xij (i G / ) .

5.4. Optimality for Submodular Flows

155

Note that (ii) of Corollary 5.2 is also expressed in a dual form as follows: Problem Ps has a feasible flow if and only if (5.37) holds for each X C V, disjoint X{ C V (i € /) and codisjoint Xij (j € Jj) (for each i E I) such that X = Uie/ Xi and X\ = CljeJt ^ij (i € /) (see the remarks after Theorem 2.6). Since the description of the algorithm for the common base problem given in Section 4.3 depends on the base polyhedron B(/) but not on the submodular function / or the system of linear inequalities expressing B(/), the algorithm also works even if / is a crossing-submodular function on a crossing family, provided that an initial base in B(/) and an oracle for exchange capacities are available. For such an / , however, finding a base in B(/) together with determining the nonemptiness of B(/) is itself a nontrivial problem. For a fixed element e € E, define families F\ = {X | e ^ X € J7} U {V}, J=2 = {X | e <E X <E 3=} U {0} and J^ = {E - X \ X € J^2}, and let / i and /2, respectively, be restrictions of / to J-\ and J-^- Then, J-\ and J-^ are intersecting families and / i and j ^ are, respectively, an intersectingsubmodular function and an intersecting-supermodular function. Since B(/) = B ( / i ) n B ( / 2 ) = B(/i) HB(/ 2 # ), a vector in B(/) is a common base in base polyhedra B(/i) and B ( / * ) , if both B(/i) and B ( / * ) are nonempty. Since P(/i) (or P(/*)) is a submodular (or supermodular) polyhedron, we can find a maximal (or minimal) vector in P(/i) (or P ( / * ) ) by adapting greedy algorithm II in Section 3.2.b. A vector in B(/), if any exists, is thus found by the algorithm for a common base problem (see [Frank84], [Puji87b]). A more direct algorithm is given by [Frank + Tardos88] (also see [Naitoh + Fuji92]) as a bi-truncation algorithm.

5.4. Optimality for Submodular Flows Consider the submodular flow problem Ps described by (5.1), where (V, f) is a submodular system on V with f(V) = 0. We show the following optimality theorem. Theorem 5.3: A submodular Bow ip: A —> R. for Problem Ps is optimal if and only if there exists a function p: V —> R such that, defining 7 p : A R by (5.38) 7P(a) = 7 (a) + p(d+a) - p(d~a) (a € A),

156

III. NEOFLOWS

we have for each a E A 7 P (a) > 0

=*


(5.39)

7p(a)<0

=?

y?(a) = c(a)

(5.40)

and such that the boundary dip: V —> R is a maximum-weight B(/) with respect to the weight function p.

base of

(Proof) The "if" part: By an elementary calculation we have

J2 l(a)
aeA

(5.41)

ueV

The "if" part immediately follows from (5.41). We postpone the proof of the "only if" part. Q.E.D. To prove the "only if" part of Theorem 5.3 we need some preliminaries. Any function p: V — R, is called a potential. A potential p satisfying the conditions of Theorem 5.3 is called an optimal potential. Given a submodular flow ip, we define the auxiliary network Afv = {Gip = ( y , ^ ) , ^ ^ ^ ) , where Gv is the graph with vertex set V and arc set Ap given by

Ap = 4 U 5 J U C , ,

(5.42)

A*v = {a | a € A, ip(a) < c(a)},

(5.43)

B* = {a | a € A, c(a) <
(a: a reorientation of a), (5.44)

CV, = {(M,U)

I u,v £ V, u £ dep(d
(5.45)

Cy. Ap —> R is the capacity function given by f c(o) - ^(a) c
(a € ^ ) _ (a € i?£, a(e ^4): a reorientation of a) (a = (u,v) £ C R is the length function given by f 7 («)_ 7ip(a) = < —7(a)

[ 0

(a € ^ ) _ (a € B^, a(e A): a reorientation of a)

(a= (u,v) eCp).

(5.47)

5.4. Optimality for Submodular Flows

157

We call the graph GQ^ = (V, Cv) the exchangeability graph associated with the base dip of B ( / ) . Also, we call a directed cycle of negative length a negative cycle. Lemma 5.4: Let ip be an optimal submodular flow for Problem P$. Then there exists no negative cycle, relative to the length function 7^, in the auxiliary network Mv. (Proof) Suppose, to the contrary, that there exists a negative cycle in Mv. Let Q be a negative cycle having the smallest number of arcs in M^ and let the arcs in Cv lying on Q be given by (ui,V{) (i € / ) . We show that by an appropriate numbering of arcs (v,i,Vi) (i € /) the assumption of Lemma 4.5 is satisfied. Suppose, to the contrary, that the assumption of Lemma 4.5 cannot be satisfied by any numbering of a r c s (ui,Vi)

(i € I),

i . e . , t h e r e a r e a r c s (uik,Vik)

t h a t ik e I (k = 1 , 2 ,

a n d (uik,Vik+1)

€ C^

(k = l,2,---,p)

such

(k = l,2,---,p)

with

ip+i = i\. Then for each k = 1, 2, ,p let Qk be the directed cycle formed by arc (uik,Vik+1) and the path in Q from vik+1 to uik (see Fig. 5.4). Since l^{{uik,vik))

=

r

rv((uik,vik+1))

= 0 (k = 1,2,

, we see t h a t

v

E >(<W = (P - qhv(Q) < 0

(5.48)

fc=l

for some integer q such that 1 < q < p, where 7¥,(Qfc) and 7¥,(Q) are the lengths of Qk and Q relative to the length function 7^. It follows from (5.48) that there is at least one Qk (k = 1,2, ) having a negative length and such a directed cycle Qk has a smaller number of arcs than Q. This contradicts the definition of Q, so that by an appropriate numbering of arcs (ui,Vi) (i € I) the assumption of Lemma 4.5 is satisfied. Define a = min{c¥,(a) | a lies on Q}

(5.49)

and modify the flow ip as

f ^{a) + a ip'{a) = < ip{a) — a ^ if{a)

(aeA^nA^Q)) (a E B^ D Alf(Q), (otherwise)

a: a reorientation of a) (5.50)

III.

158

Figure 5.4: Qk (i = 1, 2,

NEOFLOWS

,p).

for each a € A, where A ^ Q ) denotes the set of arcs lying on Q. Then
E l{a)
(5-51)

This contradicts the assumption that ip is an optimal submodular flow. Consequently, there is no negative cycle in A/^. Q.E.D. The proof technique concerning (5.48) was originated by the author [Fuji77a, 77b] for matroids and may be interesting in its own right (also see [Zimmermann82]). Now, we show the "only if" part of Theorem 5.3. (Proof of the "only if" part of Theorem 5.3) Let ip be an optimal submodular flow for Problem Pg. Then, from Lemma 5.4 there exists no negative cycle in the auxiliary network Mv relative to the length function 7^. This implies that there exists a potential p: V —> R such that 7vJ)P (a)

=

7¥ ,(a)

+ p{d+a) - p{d~a) > 0

(5.52)

5.4. Optimality for Submodular Flows

159

for each a € A^. (Such a potential p can be found by a shortest path computation from a fixed origin s outside Mv, where the origin s is connected with each vertex of V by a new arc of any finite length, say, zero.) We can easily see that (5.52) for a e i * U B * implies (5.39) and (5.40) and that (5.52) for a € Cv implies p(u) > p(y)

(u € dep(d(p,v) — {v}).

(5.53)

From (5.53) and Theorem 3.16 d(p(E B(/)) is a maximum-weight base of B(/) with respect to the weight function p. Q.E.D. It should be emphasized here that Theorem 5.3 is independent of how the base polyhedron B(/) is represented by a system of linear inequalities. [Also note that the above proof shows that a submodular flow ip is optimal if and only if there is no negative cycle in the auxiliary network Af
submodular flow ip, (i) find a negative number of arcs in the auxiliary network flow

This is the primal algorithm given in [Fuji78a] and [Zimmermann82] (also see [Cui + Fuji88] and [Zimmermann92]). When c, c, / and an initial

= A*UB*UC, A* = {a I a € A, c{a) = +00},

(5.54) (5.55)

B* = {a I a € A, c(a) = — 00} (a: a reorientation of a), (5.56) C = {(u, v) I u, v, € V, VX € V: (v £ X ^ u e X)}

(5.57)

160

III. NEOFLOWS

and 7: A —> R is tiie length function defined by (a € 4*) f 7 (a) 7(a) = < —7(0) (aeB*, a(e A): a reorientation of a) [ 0 {a (EC).

(5.58)

Suppose that there exists a submodular flow for Problem P$. Then, there exists an optimal submodular Row for Problem Ps if and only if there is no negative cycle in J\f relative to the length function 7. (Proof) The "if" part: Suppose that there is no negative cycle in J\f relative to 7. Then there is a potential p: V —> R such that for each a £ A %(a) = 7(0) + p{d+a) - P{d~a) > 0,

(5.59)

where d+ and d~ are with respect to G. (5.59) implies 7(a) +p(d+a) -p{d~a) > 0 (a £ A, c{a) = +00),

(5.60)

7(a) +p(d+a) -p(d~a)

(5.61)

< 0 (a £ A, c(a) = -00),

p{u)>p(v) ((u,v)£C).

(5.62)

Since (5.1c) is expressed as df(X)

< f(X)

(IeD),

(5.63)

d
(5.64)

where (5.64) is void, and since dtp(X) = f(A+X)

-
(5.65)

the linear-programming dual of Problem Ps with (5.1c) being replaced by (5.63) is described as Ps*:

Maximize ]T ^(a)c(a) - ]T aa)c(a) - ^ aeA

aeA

subject to |(a) - f (a)- ^{v(X)

r,(X)f(X)

(5.66a)

XeV

\X eV, a € A+X}

+ Y.MX) \xev, ae A~x} = 7 ( a ) (a G A), |(a) = 0

( a € i , c(a) = -oo),

(5.66b) (5.66c)

5.4. Optimality for Submodular Flows

161

£(a) = 0

(a € A, c(a) = +00),

(5.66d)

i,I,r)>

0,

(5.66e)

where £, £: A —> R, 77: £> —> R and we should regard (5.66a) as the objective function with the terms £,(a)c(a) (a £ A, c{a) = —00) and ^(a)c(a) (a € A, c(a) = +00) being suppressed. Because of (5.62) there is a minimal chain C: ® = SocS1C---cSn

=V

of V such that p is constant on each quotient Sk — Sfc-i (k = 1, 2, Using this chain C and the potential p, define ri(Sk)

= Pk ~ Pk+i

(k = l,2,---,n-l),

(5.67) , n).

(5.68)

where pk is the value of p taken in Sk — <Sfc-i and note that r)(Sk) > 0. Also define i](X) = 0 for other X € V. Moreover, define f(a)

=

~i{a) + p{d+a) - p{d~a)

£(a)

=

-7(a)-p(<9 + a)+p(<9~a)

(a € A, c(a) = +00),

(5.69)

(a G A, c(a) = - 0 0 ) , (5.70)

and for each arc a 6 A with c(a) < +00 and c(a) > —00 define £(a) and ^(a) such that £(a), C(a) > 0 and (5.66b) holds. We can easily see that thus defined £, ^, 77 satisfy (5.66b)~(5.66e), where note that for each arc a € A with c(a) = +00 and c(a) = —00 we have 7(0) +p(d+a) —p(d~a) = 0 due to (5.60) and (5.61). Since the dual of Problem P$ has a feasible solution and the feasibility of the primal problem Ps is assumed, there exists an optimal solution of Problem P5. The "only if" part: Suppose that there is a negative cycle in J\f relative to the length function 7, and let Q be such a negative cycle in J\f. Then for any positive a, if we define R by (5.50), ip' is feasible for Problem Ps because of the definition of J\f and we have (5.51). Since a (> 0) is arbitrary and 7
(oei),

(5.71)

162

III. NEOFLOWS d^X)(=^(A+X)-cp(A-X))
(5.72)

for the submodular flow problem Pg is totally dual integral. (Proof) Consider a submodular flow problem Pg such that the coefficients 7(a) (A € A) of the objective function are integers and that there exists an optimal submodular flow (p. From the proof of the "only if" part of Theorem 5.3 there exists an integral potential p: V — Z such that for each a £A 7 p (a) = 7(a) + p{d+a) - p(d~a) > 0 => v?(a) = c(a),

(5.73)

= 7 ( a ) + p(9+a) - p(a~a) < 0 = } <^(a) = c(a)

(5.74)

7 p (a)

and for each u EV and u € dep(d(p, u) — {u}

p(u) p2 >

(5.75)

> pi be the distinct values of p(u) (u £ V) and define

Wi = { u \ u e V , p ( u ) >

P i

}

(i = 1,2,

(5.76)

From (5.75) we have dip(Wi) =f ( W i )

(i = 1 , 2 ,

(5.77)

For the dual problem Ps* in (5.66) of F^ define £(a)

= max{0,7p(a)}

?(a)

=

max{0,- 7p (a)}

(a € A),

(5.78)

(aei),

v(Wt) = Pl-Pi+1 (i = 1,2, ??(X) = 0 for other X € V.

(5.79)

,

(5.80) (5.81)

We can easily see that these £, ^ and 77 satisfy (5.66b)~(5.66e). Moreover, from (5.73), (5.74) and (5.76)",

E K«)c(a) " E ?(«)^(«) " E »?(*)/(*) aeA

aeA

XeV l-l

= E 7P(a)v(a) - E ^ ~ Pi+O/C^i) aeA

j=l 1-1

= E 7p(a)v(a) - E f e ~ Pi+O^CWi) aeA

i=l

5.4. Optimality for Submodular Flows

163 I

= £ 7P(«M«) - E ^ ( ^ - w-i) aeA

i=l

aeA

vev

= E7(^W,

(5-82)

aeA

where Wo = 0, we put 0 x ) = 0, and note that dcp(Wi) = dip(V) = 0. Therefore, £, £ and 77 defined by (5.78)^(5.81) form an integral optimal solution of the dual problem P$*. Q.E.D. It follows from Theorems 5.6 and 2.6 that if we replace / in (5.72) by a crossing-submodular function on a crossing family, the system of (5.71) and (5.72) is totally dual integral. Corollary 5.7 [Edm+Giles77]: For a crossing-submodular function f on a crossing family T C 2V, the system of inequalities c{a)
(a G A),

dy{X)
(5.83) (5.84)

is totally dual integral. The proof of Theorem 5.6 shows the way of constructing an optimal dual solution £, £ and r\ from an optimal potential p when / is a submodular function on the distributive lattice V. When / is a crossing-submodular function on a crossing family J- C 2V with 0, V € J- and /(0) = f(V) = 0, suppose that B(/) = B(/2) for a submodular system (V, f^) (see Theorem 2.6) and that we are given an optimal submodular flow if and an optimal potential p. Also suppose that for each Wi in (5.77) we have the expression of f^iWi) in terms of / as follows (see Theorem 2.6). /2(Wi) = ] T Y, f(xm)

(i = l,2,---,l-l),

(5.85)

j£.Ji keKij

where V — (Ufcex -^-ijk) (j € Ji) form a partition of V — Wi for each i = 1, 2, / — 1 and Xjj^ (k G ii'jj) are disjoint sets (as subsets of V") in

164

III. NEOFLOWS

F for each i = 1,2, — 1 and j 6 J\. Here, note that f2{V) = 0. Then define for each X G T r,(X) = Y,{pi - Pi+i I x = xijk,

i G {l,

, i - l}, j G Jt, k G i ^ } , (5.86) where the summation over the empty set is equal to zero. (Note that if all the X^k a r e distinct, we have rj(Xijk) = Pi — Pi+i (i € {1, / — 1}, j € J2, A; G ify) and r?(X) = 0 for other X G F.) | , f defined by (5.78) and (5.79) and this r) form an optimal dual solution, since (5.82) with T> replaced by T holds. , We show the way of finding an expression in (5.85) for each i = 1, 2, I — 1. Put x = dip and define F(x) = {X | X € F, x(X) = f(X)}.

(5.87)

We assume an oracle which, for each ordered pair (u, v) of distinct vertices u, v G V, gives a u-v cut X G ^"(x) or answers that there is no u-v cut X € F(x), where a it-t; cui is a set X C V such that it € X and v ^ X. Let G = (V, A(x)) be the exchangeability graph associated with x, i.e., A(x) = {(v,u)

| t / 6 7, i ) € d e p ( s , u ) - { u } } ,

(5.88)

where note that (v, u) E A{x) if and only if u ^ v and there is no u-v cut X G T{x). Choose any W-i (i = 1, 2, , I — 1). Let Gj be the graph obtained from G by deleting all the vertices in W{ together with the arcs incident to Wi, and let Uij (j G Jj) be the vertex sets of the connected components of G{. Note that A+Uij = 0

(j E Ji)

(5.89)

in G, so that f2(V - Uij) = x(V - Uij) (jeJi).

(5.90)

Therefore, for each u G V — Uij and v G Uij there exists a it-?; cut X € T(x). If X Pi Uij ^ 0, choose v' € X fl t/jj such that for some w € C/j^ — X we have (v1, w) G j4(a;). Such a i/ exists since C/jj — X / 0 and C/jj is the vertex set of a connected component of G\. There exists a u-v' cut X' € J~(x) and either X and X ' cross or I ' C I , due to the way of choosing v'. (Note that u <E X D X', v' E X - X1 and w <£ X U X'.) Since F{x) is a crossing family, we have I f l l ' e ^"(s) a n d u e l n l ' c l . P u t l ^ l n X'.

5.4. Optimality for Submodular Flows

165

If X n Uij ^ 0, then repeat this process until we have X f) Uij = 0. After repeating this process at most \U{j\ times we have X £ T{x) such that u E X C V - Uij. Denote this I by I t t . In this way, for each u £ V — Uij we can find Xu such that u € X u C V - Uij. Consider the hypergraph H = (V - Uij,{Xu | u E V - Uij}) and let X^ (k £ Kij) be the vertex sets of the connected components of H. Since J~(x) is a crossing family of subsets of V, we have X^ E J~(x) (k £ K^). Consequently,

/2(Wi) = x(Wi) ="£ E

x(Xijk)-(\Ji\-l)x(V)

= E E /(*«*)

(5-91)

jeJi k€Kij

since x(F) = 0. When / is an intersecting-submodular function on an intersecting family f C 2 y with 0, V € T and /(0) = f(V) = 0, for each u & Wi we can find X u € T{x) such that u € Xu C Wj. (Note that for any v, v' £ V — Wi there exist a u-v cut X € J~(x) and a u-v' cut X' E J~(x) and that u E X PI X ' € ^"(x) since ^"(x) is an intersecting family.) Let JQj (j £ Ji) be the vertex sets of the connected components of the hypergraph H = (Wi, {Xu | u E Wi}). Then X{j E f(x) and we have

/iW)=E/(Xu),

(5-92)

where / i is the submodular function appearing in (i) of Theorem 2.6. Let G = (V, A) be a connected (directed) graph. In Section 3.3.b we considered strongly /c-connected reorientations of G for a positive integer k. For the crossing-submodular function K^: 2V — R denned by (3.105), the problem of finding a minimum-cost strongly /c-connected reorientation (p: A —> {0,1} is described in a relaxed form as p(k).

Minimize ^ 7(a)v?(a)

(5.93a)

aeA

subject to 0 < ip(a) < 1 (a € A), dtp{U) < K^(U) (U C F ) ,

(5.93b) (5.93c)

where 7: A —> R is a cost function and ip: A —> R is not restricted to integral vectors. Due to Corollary 5.7 the polyhedron of feasible flows ip is

166

III. NEOFLOWS

integral and there exists an integral optimal solution, if any optimal solution exists, which is a minimum-cost strongly /c-connected reorientation of G. Problem (5.93) is a typical example of the submodular flow problem (see [Frank80,81c]). Note that Problem (5.93) with k = 1 has a feasible flow if and only if G is connected and has no bridge. For a connected graph G = (V, A) let T>(G) be the set of all the ideals of G, i.e., V(G) = {W | W C V, A+W = 0}. (5.94) For each W € V{G) - {0, V} we call A~W(^ 0) a directed cut. A subset B of arc set A is called a directed-cut covering of G if B has nonempty intersection with each directed cut of G. C. L. Lucchesi and D. H. Younger [Lucchesi + Younger78] consider the problem of finding a minimum directed-cut covering, which is formulated in a relaxed form as DCC: Minimize ^ v(a)

(5.95a)

subject to 0 < tp(a) < 1 (a E A), dip(W) < K{1)(W)

(5.95b)

(W € T>(G)), (5.95c)

where K^ is defined by (3.105) with k = 1. It should be noted that K^ is a crossing-submodular function on T>(G). The linear-programming dual of DCC is given by DCC*: Maximize - ^ ^(a) aeA

^

T]{W)KW{W)

(5.96a)

WeV(G)

subject to - f (a) + ^{r]{W)

\ W € V{G),a € A"W} < 1 (a <E 4),

£, rj > 0.

(5.96b) (5.96c)

Therefore, Problem DCC* becomes Maximize ^ min{0, [1 - X X ^ ^ 7 ) I ^ +

^ rj(W) weo(G)-{0,y} subject to r? > 0.

€ Z) G

( ')) a

e

A

~^}]} (5.97a) (5.97b)

We see from (5.95)~(5.97) and Corollary 5.7 that there exists an integral optimal solution rj of dual DCC* such that (1) i](W) € {0,1} (W €

5.5. Algorithms for Neoflows

167

V(G) - {0,F}) and (2) directed cuts A~W with n(W) = 1 are disjoint. Consequently, we have the following theorem. Theorem 5.8 [Lucchesi+Younger78]: The minimum cardinality of a directed-cut covering ofG is equal to the maximum cardinality of a set of disjoint directed cuts of G.

5.5. Algorithms for Neoflows We consider maximum neoflow and minimum-cost neoflow problems and show algorithms for these problems. (a) Maximum independent flows The maximum flow problem for neoflows seems to be easily formulated by the independent flow problem. The maximum independent flow problem is: PMV- Maximize dip(S+)

(5.98a)

subject to c(a) < ip(a) < c{a) (a € ,4),

(5.98b)

dtp(v) = 0 (v(EV-(S+US-)), S+

(d^)

€ P(/+),

(5.98C)

(5.98d)

- (dtpf- € P ( / " ) ,

(5.98e)

where we consider the same network described in Section 5.1.b. We denote (dcp)s and —(dtp)s by d+tp and d~ip, respectively, in the following. We suppose there exists a feasible flow. A feasible flow, if any exists, can be found by adapting the algorithm in Section 4.3 as discussed in Section 5.3 for submodular flows. Given a feasible flow ip, we define an auxiliary network A/^ = (G v = (V U {s + , s~}, A^), cv, s + , s~) with source s + and sink s~ as follows. Gv is the underlying graph with vertex set V U {s+, s~} and arc set A^ defined by Av

= S+US-l>A+UA-UA*vl>B;, +

+

+

(5.99) +

S+ = {(s ,v)

veS

-s&t (d f)},

S- = {(v,s~)

v£S~ -s&t-(d-
+

A+ = {(u,v) | v € sat (d tp),

(5.100) (5.101) +

+

u € dep (d tp,v)

- {v}}, (5.102)

III. NEOFLOWS

168

Ay = {(v,u) | v € sat (d cp), u e d e p (d
(5.104)

Bv = {a a<EA, ip(a) > c(a)},

(5.105)

where a denotes a reorientation of a, sat"1" (sat~) is the saturation function with respect to (Z> + ,/ + ) ((£>",/")) and dep + (dep~) is the dependence function with respect to (V+, / + ) ((T>~, / " ) ) . Also, c^: A^ —> R is denned by

vW

c+(d+
(a = (s+,v)eS+) (a = (v,s~) € S~) (a=(u,v)eA+)

] c-(d-
(a € Ay) (a € B* a(G A): a reorientation of a), (5.106) where 6"1" (or c~) denotes the saturation capacity associated with (P"*~, / + ) (or (T>~, / " ) ) and c + (or c~) the exchange capacity associated with (T>+, f+) (or(p-,/-)). We suppose that a feasible flow (an independent flow)

Input: an independent flow ip in network J\f = (G = (V, A; S+, S~), c, c, (X>+, /+), (V~, / " ) ) and a fixed numbering vr: V -> {1, 2, , \V\}. Output: a maximum independent flow ip in J\f. Step 1: While there exists a directed path from s+ to s~ in the auxiliary network Afy = (Gy = (V U {s + , s~}, Ay),Cy, s+, s~), do the following. (*) Find a lexicographically shortest path P from s+ to s~ in My and put (5.107) a <— min{c¥,(a) | a: an arc lying on P } ,

^V;

\<^(a)-a ( o e B J n i ^ P ) ,

v

a: a reorientation of a(€ A)).

;

5.5. Algorithms for NeoRows

169

(End) In this algorithm a lexicographically shortest path from s+ to s~ in Afp is a directed path from s+ to s~ in Mv which has the minimum number of arcs among directed paths from s + to s~ in J\f^ and whose vertex sequence (s + , Vi, vp, s~), say, gives the lexicographically minimum se+ quence (TT(VI), TT(VP)) among directed paths from s to s~ in Nv having the minimum number of arcs. Also, AV(P) is the set of arcs, in Av, lying on P. The validity of the above algorithm can be shown almost in the same manner as in the proof of the validity of the algorithm for the intersection problem in Sections 4.1.b and 4.I.e. The algorithm described above finds a maximum independent flow after repeating (*) at most |V^|3 times. The maximum independent flow problem naturally includes the intersection problem, where the underlying graph is the bipartite graph representing the bijection between S+ and S~. Theorem 5.9 (The maximum-independent-flow minimum-cut theorem) (cf. [McDiarmid75], [Fuji78a]): Suppose that the maximum independent How problem PMI hasa feasible Row. Then we have max{d(p(S+) \ tp is an independent flow (satisfying (5.98b)~(5.98e))} = m i n { / + ( 5 + -U) + c(A+U)

- c(A~U)

+ f-(S~nU)\UQ

V}, (5.109)

where A + and A~ are with respect to G and we regard f+(X) = +oo

(f-{Y) = +oo)ifX<£V+ (Y<£V~). Moreover, if c, c, / + and f~ are integer-valued and Problem PMI is feasible, then there exists an integral maximum independent Row for Problem PMI(Proof) Let ip* be a maximum independent flow for Problem PMI-,aiiUconsider the auxiliary network Mv* associated with ip*. Let U* be the set of vertices in V which are reachable by directed paths from s+ in the auxiliary network M^*. Then we have S+-U* C s a t + ( d V ) ,

(5.110)

S~nU* C sat'(d'tp*)

(5.111)

170

III. NEOFLOWS

since there is no directed path from s + to s~ in Nv*. Hence, by the definition of U*, dep+(d+
(v£S~nU*),

(5.113)

tp*(a) = c(a)

+

(aeA U*),

(5.114)

tp*(a) = c(a)

(aeA~U*).

(5.115)

From (5.112) and (5.113), d+
= f+(S+

-U*),

d-(p*(S-nu*) = f-(S~nu*)

(5.116) (5.117)

due to Lemma 2.1. From (5.114) and (5.115),
tp*(A-U*) = c(A-U*).

(5.118)

It follows from (5.116)^(5.118) that dp*(S+) = d+p*(S+ - U*) + p*(A+U*) -
5.5. Algorithms for Neoflows

171

Now, consider a supermodular system (T>+ ,g+) on S+ instead of submodular system (T>+, / + ) and also consider the following system of inequalities. c(a) < (p(a) < c{a) (a € A), (5.121) d
€ P(/-),

where P(g + ) is the supermodular polyhedron associated with Then we have the following theorem.

(5.122) (5.123) (5.124) (V+,g+).

Theorem 5.10: There exists a feasible flow cp satisfying (5.121) — (5.124) if and only if we have for each U C V such that S+nU € V+ and S~C\U € Z>~ g+(S+ DU)- f-(S~ DU)< c(A+U) - c(A~U)

(5.125)

and for each U C V such that S+ U S~ C U 0
(5.126)

Moreover, if there exists a feasible Bow and c, c, g+ and f~ are integervalued, then there exists an integral feasible Bow. (Proof) Define V C 2V and g: V -> R by

p = {[/ | [/ c v, S+ nu ev+, s~ n t/ e 2?~}, rrn =

51

}

/ g+(s+ nu)-f~(s-nu)

(uev, (5+ u

[0

(f/eD, 5+ur c[/).

(5.127)

0) (5.128)

If there is a feasible flow, we must have

g+(S+)-f-(S-)<0.

(5.129)

Also, (5.125) with U = V implies (5.129). Therefore, we assume (5.129). Due to (5.129), the function g: V — R denned by (5.128) is a supermodular function on the distributive lattice V with 0, V € V and #(0) = #(F) = 0. We have x € B(g) if and only if xS+ € P(ff+), - x S " € P ( / ~ ) , x y - ( S + u ^ } = 0,

(5.130)

172

III. NEOFLOWS x(V) = O.

(5.131)

It follows from (2.65) and Theorem 4.13 that there exists a feasible flow satisfying (5.121)^(5.124) if and only if g(U)
(U e V).

(5.132)

We see that (5.132) is equivalent to (5.125) and (5.126), under condition (5.129). The integrality part of the present theorem follows from the counterpart of Theorem 4.13. Q.E.D. Theorem 5.10, where c = 0, c > 0, / ~ is a polymatroid rank function and g+ is the dual supermodular function of a polymatroid rank function, is shown in [Fuji78d]. The discrete separation theorem also follows from Theorem 5.10. The readers may deduce Theorem 5.10 from the feasibility theorem for submodular flows (Theorem 5.1).

(b) Maximum submodular flows For the submodular flow problem with the network Ms = (G = (V,A),c, c, 7, (T>, / ) ) , disregarding the cost function 7, let us consider a "max-flow" problem PMS given as follows [Cunningham + Frank85]. Let ao be a fixed reference arc in A. PMS'- Maximize ip(ao)

(5.133a)

subject to c(a) < cp(a) < c(a)

(a E A),

dip£B(f).

(5.133b) (5.133c)

We assume that there is a feasible flow in Ms and define the auxiliary network Mv = {G^ = (V, Ay),^) associated with ip as follows. Gv is the graph with vertex set V and arc set A^ denned by (5.42)~(5.45) except that B^p = {a I a € A — {ao}, c(a) < f(a)}

(a: a reorientation of a) (5.134)

and Cy,: A^ —> R is denned by (5.46).

An algorithm for finding a maximum submodular flow

5.5. Algorithms for Neoflows

173

Input: a feasible flow ip in A/5 with reference arc ao and a vertex numbering 7r: y —> {1, 2, , |y|} which defines the lexicographic ordering among directed paths from d~ao to d+ao in A/"^. Output: a maximum submodular flow ip in A/5 with reference arc aoStep 1: While
^

l j

f ^(a) + a (aG^n^(Q)) 1
where if a = +00, then stop (the flow value ip(ao) can be made arbitrarily large). (End) Here, A(p(Q) is the set of the arcs in A^ lying on Q and the lexicographically shortest path P is the directed path from d~ao to d+ao in A/^ which has the minimum number of arcs among directed paths from d~ao to d+ao in M^ and whose vertex sequence (d~ao, v\, vp, d+ao), say, gives the lexicographically minimum sequence (vr(ui), , vr(wp)) among directed + paths from d~ao to d ao in J\fv having the minimum number of arcs. The algorithm terminates after repeating (*) O(|y| 3 ) times. The analysis is by the same technique as shown in Section 4.I.e. Theorem 5.11: For the maximum submodular Bow problem PMS described by (5.133), max{ip(ao) \ (p is a feasible flow in A/5} = min{c(a 0 ), mm{c(A~X) - c(A+X - {a0}) +f(X)

I X € V, a0 € A+X}}.

(5.135)

Moreover, if c, c and f are integer-valued and there exists a maximum submodular Row, then there exists an integral maximum submodular Bow in A/5.

174

III. NEOFLOWS

(Proof) The maximum flow value is equal to the maximum value of C{CLQ) under the constraint that the network Ms has a feasible flow, where c(ao) is regarded as a variable. Therefore, relation (5.135) is deduced from Theorem 5.1. The integrality property also follows from Theorem 5.1. Q.E.D. Theorem 5.11 can also be shown algorithmically. When the algorithm terminates with a maximum flow ip* such that f*(ao) < c(ao), let U be the set of the vertices which are reachable by directed paths from d~ao in the then obtained Mv*. From the assumption, d~a,Q E U and d+ao (£ U. Define W = V — U. It follows from the definition of M^* that (p*(a) = c(a) (oGA-W),

(5.136)

V*{a) = c{a) (a e A+W - {a0}),

(5.137)

dep(d
(5.138)

where A + and A~ are with respect to G. From (5.138) we have dy*{W) = f{W).

(5.139)

and from (5.136) and (5.137) d
(5.140)

Combining (5.139) with (5.140), we get ^*(a0) = c(A-W) - c(A+W - {a0}) + f(W).

(5.141)

On the other hand, we can easily see that for any feasible flow

with a^ E A+X,
(5.142)

From (5.141) and (5.142) we have (5.135), where the case when
5.5. Algorithms for NeoRows

175

When f*(ao) < c(ao), the above denned U(= V — W) is called a minimum cut in Ms with reference arc aoWe can also consider the minimum submodular flow problem which is to minimize
(c) Minimum-cost submodular flows As a minimum-cost flow problem for neoflows, consider the submodular flow problem Ps, described by (5.1), in network Ms = {G = (V, A), c, c, 7, (T>, / ) ) . r

Ps'- Minimize ^

){a)ip{a)

(5.1a)

a€A

subject to c(a) < tp(a) < c(a) (a £ i ) , dfEB(f),

(5.1b) (5.1c)

where (V, / ) is a submodular system on V, the vertex set of the underlying graph G = (V, A). We suppose that Problem Ps has a feasible flow. We shall show an algorithm which tries to find a feasible flow ip: A —> R and a potential p: V —> R satisfying the optimality condition of Theorem 5.3. Suppose that we are given a feasible flow ip in J\fs- Choose a potential p: V —> R such that dip € B(/) is a maximum-weight base of B(/) with respect to the weight function p, i.e., p{u) > p{v) for each u, v (E V with u G dep(dip,v) — {v}. For example, p = 0 (the zero function) satisfies this requirement. We define the auxiliary network Afv,p = (Gv = (V, A^,),^, 7^)P) as follows. Gv is the graph with vertex set V and arc set Av defined by (5.42)~(5.45) and c^: Av -^ R is defined by (5.46). We define 7 ^ : Av -^ R by ^Aa)=Ma)+P(d+a)-p(d~a) (aeA^), (5.143) where 7^ is given by (5.47). Now, an algorithm based on Cunningham and Frank's [Cunningham+ Frank85] is given as follows. Also, compare it with an out-of-kilter method given in [Fuji87b].

176

III.

NEOFLOWS

An algorithm for finding an optimal submodular flow Input: a feasible flow ip in Ms and a potential p: V —> R such that p(u) > p{v) {v
-

v;

I c(a)

(a € A,

7p (a)

v

=0),

c°(a) = { ^ ) f a e t 7 p f a ^ m

;

(5-145)

v ;

v ; [ c{a) (a € A, 7 p (a) = 0) + + (where 7 p (a) = 7(0) + p ( 9 a ) — p(<9 a) with 9 and d being defined with respect to G) and, letting p\ > P2 > > pt be the distinct values of p(v) (v € V") and defining

Si = { v \ v £ V , p ( v ) >

P i

}

(i = 1,2,

,

5 0 = 0,

(5.146) (5.147)

(T>°, f°) is the submodular system given by the direct sum ^(P,/)-^/^! of the set minors (£>, / ) Si/Si-!

(i = 1, 2,

(5.148) , £).

If the maximum flow value is equal to +00, then stop (Problem Ps does not have a finite optimal solution). Otherwise put ip <—
5.5. Algorithms for NeoRows

177

If (p°(a0) < c(a°), then for the auxiliary network J\f°0 associated with the current submodular flow ip° in J\f° let U be the set of the vertices which are reachable by directed paths from d~aP in A/"°o, define Hi H2

= {a | a € A~U, 7 p (a) < 0}, +

= {a | a € A U,

7 p (a) >

(5.149)

0},

(5.150)

+

(A , A " in (5.149) and (5.150) are with respect to G.) H3 = {(u,v) | u€ U, «£ V-U,

u£dep(dip,v)- {v}},(5.151)

(dep is with respect to (T>, /).) p*

= min{min{|7p(a)| | a € Hi U H2}, mm{p(u) -p(v) I (u,v) G F 3 } } ,

(5.152)

and put p(u)

+- p(u) -p*

(ue U).

(5.153)

(1-2) If a 0 ^ A, let a 0 € A be the reorientation of a 0 and, starting with (p, find a minimum submodular flow ip°, with reference arc a 0 , in the modified network J\f° as defined in Step (1-1). Carry out Step (1-1) mutatis mutandis. (End) It should be noted that when we carry out (5.149)^(5.153), we have since a0 e -Hi, and that p* in (5.152) is a finite positive HiUH2UH3^(d number. For a feasible flow

0 and c(a) < (fi(a) or (ii) 7p(a) < 0 and
178

III. NEOFLOWS

(4) ip is a submodular flow in Ms and dip is a maximum-weight base of B(/) with respect to weight function p. Therefore, when the cost function 7 is integer-valued, the algorithm terminates after at most ]CaeA W*2)! maximum and minimum submodular-flow computations in Step (1-1) and Step (1-2) if we start with the initial potential p = 0. Moreover, for any real cost function 7 the algorithm also terminates after finitely many steps, as shown below. While the inner cycle (*) of Step (1-1) is repeated, potential p(d+a°) remains fixed and p(d~a°) is decreased by (5.153), where note that for each (u, v) £ H3 we have p(u) > p(v). Every time we change the potential p by (5.153), one of the following two cases occurs: (i) at least one arc in H\ U H2 becomes in kilter, (ii) the set of the vertices which are reachable from d~aP in the new auxiliary network J\f°0 is augmented. Note that the total number of the occurrences of Case (i) is at most \A\ and that Case (ii) occurs consecutively at most |V| — 1 times without increasing ip°(a°). While 0 for all a € A,p, then the obtained ip is an optimal submodular flow and p is an optimal potential due to Theorem 5.3. If the maximum flow value for the modified network M° in Step (1-1) is equal to +00, then there exists a directed cycle Q, containing a0, in the auxiliary network Af® = (G% = (F, A^,), cjl) associated with the current ip such that Ctp(a) = +00 for any arc a in Q. By the definition of jV°, Q is also a directed cycle in the auxiliary network J\fv = (Gv, 0^,7^) for the original submodular flow problem P5 and the length of Q relative to the length

5.5. Algorithms for NeoRows

179

function 7^ is negative. Moreover, ip remains to be a (feasible) submodular flow if we change

—00 for each a £ A. ^t.»j

v

i_n_,vj.

i

KJI. djv^j.v^

J.WJ.

vjAVjiiaii£.v.

A cost-scaling algorithm Input: a feasible flow ip in A/5, a potential p = 0, and C = max{|7(a) | | a € A}, where the cost function 7 is integer-valued and C > 0. Output: an optimal submodular flow ip and an optimal potential p. Step 0: Discern whether Problem Ps does not have a finite optimal solution. If not, stop; otherwise put k <— |~log2C~|. Step 1: While k > 0, do the following: (1-1) For each a G A put 7(0) <- |~7(a)/2fe~|. (1-2) By the pseudopolynomial algorithm given above, starting from the current flow ip and potential p, find an optimal submodular flow ip* and an optimal potential p* in the network J\fs = (G = (V, A),c,c, 7, (1-3) Put ip <- ip*, p <- 2p*, and k <- k - 1. (End)

180

III. NEOFLOWS

Step 0 can be carried out by a shortest path computation based on Theorem 5.5. In Step (1-2) we have

I>( a )l< 2 l^l

E

(5.154)

aeA°(ip,p)

for the current 7 and p. Hence each Step (1-2) requires at most 2\A\ maximum and minimum submodular-flow computations. It follows that the cost-scaling algorithm requires 0(|.A|log 2 C) maximum and minimum submodular flow computations and that the algorithm runs in polynomial time if an oracle for exchange capacities is available. It should also be noted that, since in Step 1 we have 7(0) x 2k > 7(0)

(a € A),

(5.155)

and since Problem P$ is bounded when we proceed to Step 1, the problem on network J\fs = (G = (V, A), c, c, 7, (2?,/)) with approximated cost function 7 is also bounded due to the assumption that c(a) > —00 (a € A) (recall Theorem 5.5). If we apply the cost-rounding and tree-projection technique of the author [Fuji86] (which is an improved version of Tardos's algorithm [Tardos 85]) for the ordinary minimum-cost flows, we can get a strongly polynomial algorithm for the submodular flow problem ([Fuji + Rock + Zimmermann89]). The first strongly polynomial algorithm for the submodular flow problem was given by Frank and Tardos [Frank + Tardos85] by the use of the simultaneous approximation algorithm of [Lenstra + Lenstra + Lovasz82]. To give a strongly polynomial algorithm of [Fuji + Rock + Zimmermann89] we need a few preliminaries. For simplicity we assume that c and c take on only finite values, which guarantees the boundedness of Problem PsFor a cost function 7: A —> R and a positive real a, 7': A —> R is called an a-approximation of 7 if \y'(a) — 7(0) | < a for all a € A. The following lemma is fundamental. Lemma 5.12: Let 7' be an a-approximation of 7 and p' be an optimal potential in A/7 = (G = (V, A),c, c, 7', (T>, / ) ) . Then there exists an optimal potential p in J\f = (G, c, c, 7, (V, /)) such that we have for each u, v € V \{p'(u)-p'(v))

- (p(u)-p(v))\

< a\V\.

(5.156)

5.5. Algorithms for Neoflows

181

(Proof) Without loss of generality we assume 7'(a) > 7(0) for all a € A (by possible reorientations of arcs). If 7' 7^ 7, let v° be a vertex in V such that 7 ; (a) > 7(0) for an arc a € <5~i>°. Define -v \ / i(a) 7(a) = < ,\ \

'v

y

I 7'(a)

( a € 5"w°) ) . '

,_1KV s (5.157)

nx

K

(a € A-S~v°).

'

Starting from an optimal submodular flow ip' and an optimal potential p' in J\f' = (G,cJ~c,'j'J ( P , / ) ) , we apply a slightly modified version of the (non-scaling) algorithm for submodular flows to J\f = (G,c, c, 7, (T>, / ) ) . The modification consists only in postponing potential changings as long as possible, which does not affect the finiteness of the algorithm. Note that initially for any out-of-kilter arc a e i w e have a € S~v°,

(5.158)

0>7p'(a)>-a,

(5.159)

ip'(a) < c(a).

(5.160)

In the original version of the algorithm an out-of-kilter arc a 0 G 5~v° is chosen and (p'(a°) is increased by solving a maximum submodular flow problem with reference arc a0, which is followed by a potential changing. In the modified version we repeatedly solve maximum submodular flow problems, each with an out-of-kilter arc a € 5~v° as a reference arc, without any potential changings. If there exists at least one out-of-kilter arc after this process, then let U be the set of the vertices reachable from v° in the current auxiliary network and change the potential p by (5.149)^(5.153). Repeat this process until all the out-of-kilter arcs become in kilter. (This is repeated finitely many times.) Let p be the resultant potential and let a0 be one of the arcs which became in kilter at the last potential-changing. We can see from (5.159) that m&x{\(p'(u) -p'(v))

- (p(u) -p(v))\

< m&x{\p(u) -p{u)\

I u € V}

I

u,v(EV}

< \%'(a°)\ < a, where note that 0 < p'(u) — p(u) for all u £ V due to (5.153).

(5.161)

182

III. NEOFLOWS

Putting p' «— p and 7' <— 7 and repeating the above argument for other vertices, we see that the total change of any potential difference is bounded by a\V\. Q.E.D. From this lemma we have Theorem 5.13: Let 7' be an a-approximation of 7 and p' be an optimal potential in M' = (G = (V, A), c, c, 7', (D,f)). Then for any optimal submodular Bow (p in N= (G = (V, A),c,c,j, (V, / ) ) , (1) Va G A: ^p,(a) > a\V\ => ip(a) = c{a), (2) Va € A: lp,{a) < -a\V\ = >
> a\V\ ^> v (£ dep(dtp,u).

(Proof) The present theorem follows from Lemma 5.12 and Theorem 5.3. Q.E.D. Theorem 5.13 gives a basis for a strongly polynomial algorithm for submodular flows. We show a strongly polynomial algorithm which consists of the repeated applications of a procedure called Fundamental Cycle. Fundamental Cycle Input: Lower and upper capacity functions c and c; a partition W = {W^ I ) G 1} of V; a representation of S = ©je/S* as a direct sum of submodular systems Sl = (T>\ f) on W{ (i £ / ) ; a graph H = {V, D) with connected components Hl = (W\ El) (i 6 /) which are strongly connected; and a set J4° = {a I a € A, c{a) = c(a)} of all tight arcs. (Comment: At the initial application of this procedure we put / = {1}, W1 = V and H = (V, D) with D = {(u, v) I u, v G V, u v}.) Output: A nonnegative real M, and if M ^ 0, modified c, c, W, S, H and A0. (Comment: When M = 0, the set of all the submodular flows in the current network M = (G = (V, A), c, c, 7, S) is exactly the set of all the optimal submodular flows in the original network. When M 7^ 0, c, c, W, if and ^4° have been modified in such a way that all the input characteristics are maintained and that at least one of the following two properties holds:

5.5. Algorithms for NeoRows

183

(a) A0 is strictly larger than the input A0, (b) D is strictly smaller than the input D.) Step 1 [Construction of a potential]. (1-1) Shrink the vertex subsets Wi (i E I) of G = (V,A) into (pseudo-) vertices wl (i G /) and denote the resultant graph by G//W. Denote by G = (V, A) the graph obtained from G//W by deleting all the arcs of A0. Find a spanning forest T of G which is composed of rooted spanning trees of the connected components of G. (1-2) Put p(v) <— 0 for all roots v of spanning trees in T and construct the potential p: V —> R in such a way that 7p(a) = 7(a) + p(d+a) — p(d~a) = 0 for all the arcs a of T. (1-3) Define the potential p: V —> R by p(v) = p{wl)

(v EW\ iel).

(5.162)

Step 2 [Rounding the cost function]. (2-1) Put M <-max{|7 P (a)| | a € A - A0}.

(5.163)

If M = 0, then stop. (2-2) Put ^p(a)^lp(a)-\V\2/M 7»W7!(a)J

(a<=A), (a€A).

(Comment: \-\ means rounding to the nearest integer.) Step 3 [Solving the approximate problem]. Find an optimal submodular flow ip' and an optimal potential p' in the network N' = (G = (V, A), c, c, 7', S) by the (cost-scaling) algorithm of Cunningham and Frank. Step 4 [Extracting information on optimal submodular flows]. (4-1) For each arc a € A - A0, (i) if (7pV(a) < ~l\V\, then put c(a) <- c{a) and A0 <- A0 U {a},

184

III.

NEOFLOWS

(ii) if (7p)p'(a) > \\V\, then put c(a) <- c(a) and A0 <- ^4° U {a}, where ( 7 | V ( a ) = 7|(a) + J>'(<9+a) - p'(0-a). (4-2) For each arc («, u) € £> of the graph H, i(p'(v) —p'(u) > ^\V\, then delete (u, v) from D and from E'j to which (u, v) belongs. (4-3) For each i € I do the following (4-3-l)-(4-3-3): (4-3-1) Find a maximal chain 0 = Ul0 C C/J C C U^ = Wi of upper ideals of H%. {Comment: An upper ideal of a graph is a vertex set which no arcs enter.) (4-3-2) Define Sis(=(V*%r°))

= (V\f)-Ul/Ul_1

Wi'=U%8-Uts_l ls

(Comment:

W

(s = (s = 1,2,

(s = 1,2,

,ki)

.

a r e t h e v e r t e x s e t s of t h e

strongly connected components of Hl.) (4-3-3) Delete from the graph H all the arcs connecting distinct subsets Wis (s — 1 2 k) (4.4) p u t I

<-

S

<- ©ie/Si,

W

<-

{i8 \s =

{Wl

i€

I},

|ie/}.

(End) To find an optimal submodular flow in the original network we repeatedly apply the procedure, Fundamental Cycle. We show the validity and the strong polynomiality of this algorithm. At any stage of the algorithm the input to Fundamental Cycle is referred to as the current network with the current capacity functions, the current submodular systems, etc. Theorem 5.14: At any stage of the algorithm the following are valid.

statements

5.5. Algorithms for Neoflows

185

(1) For any optimal submodular flow ip in the current network the exchangeability graph GQ^ associated with base dip of the current submodular system is a subgraph of the current graph H = (V, D). (2) The set of all the optimal submodular flows in the original network coincides with the set of all the optimal submodular Bows in the current network. (3) While M ^ 0, each application of Fundamental Cycle strictly enlarges the set A0 of the tight arcs or strictly reduces the set D. (Proof) We show the present theorem by induction on the number of applications of Fundamental Cycle. Before the initial application of Fundamental Cycle the current network and the original network coincide and the current graph H contains all arcs (u,v) with u, v £ V and u / v. Obviously, the statements of the theorem are valid. Now, let us assume that the statements are valid before an application of Fundamental Cycle. We show their validity after this application. Suppose M ^ 0; otherwise the algorithm terminates without modifying the input and we are done. Let a* be an arc which maximizes (5.163) in Step (2-1). If we add the arc a* to the forest T in G = (V, A) (see Step (1-1)), then a unique cycle Q* is created. Let Q* be represented by a sequence of arcs as Q* = (a0, a1, , a') with a0 = a*. We assume that a* is positively oriented in the cycle Q*. Since the (pseudo-)vertices wl correspond to the shrunken strongly connected graphs Hl = (Wl,El) (i £ I), we can extend Q* to a cycle C in the graph (V, Al) D) such that all the arcs from C C\D are positively oriented. We define s

( a ) = l{a) \V\2/M

(a E A),

(5.164)

s 2 1 p(a)=lp(a)-\V\ /M

(aGA),

(5.165)

p*(v) = p(v) \VfjM

(v € V),

(5.166)

7

where we have 7

»

= 7S(«) + Ps(d+a) - ps(d~a)

(aGA).

(5.167)

We extend the domain of 7* to AliD by defining 7^(a) = 0 (a € D). (Note that this is equivalent to denning j(a) = 0 (a £ D).)

186

III. NEOFLOWS

In the following we assume that we have 7p(a*) = — M in Step (2-1). (The case when 7P(a*) = M can be treated similarly.) Then the length of the cycle C relative to the length function 7* is given by 7|(C)

= -\V\\

(5.168)

since 7p(a*) = — \V\2 and 7p(a) = 0 for any other arcs a in C. In Step 3 of Fundamental Cycle we construct an optimal submodular flow
< -\V\,

(5.169)

where ec(a) is equal to 1 or —1 according as a is positively oriented or negatively oriented in C. Since ec(a) = 1 if a € D, either (i) |(7p)p'(a)| > if a = (u,v) € D. if a € A-A0, or (ii) p'(v) -p'(u) > Since 7' is a ^-approximation of 7^ (see Step (2-2)), it follows from Theorem 5.13 that for any optimal submodular flow ip in Afs = (G, c, c, 7*, S) (1) if a e A - A0, then V

^

\ c(a)

if (T|V(«) < - | | V | ,

V

'

(2) if a = (u, v) € -D, then p'(v) - p'(u) > \ \V\ and u(£dep(dtp,v),

(5.171)

where the dependence function is defined with respect to the current base polyhedron. Therefore, in Steps (4-1) and (4-2) A0 is strictly enlarged or D is strictly reduced, which shows statement (3) in the present theorem. Moreover, we have

(|F| 2 /M)-^ 7 (a)^(a) = ^ 7 S ( « M « ) aeA

aeA

= J2 Op(«M«) - J2 P(v)d
v£V

= ^ 7 > M a ) - E ^ W ^ ) > (5-172) aeA

iel

5.5. Algorithms for NeoBows

187

where wl is the (pseudo-) vertex representing the shrunken set Wl and p(v) = p{wl) 0 e W\ i € /) due to Step (1-2). Since dipiW*) (i e /) are independent of the choice of dip € B = © j € / B \ where B l is the base polyhedron of submodular system S* for each i € /, it follows from (5.172) that the set of all the optimal submodular flows in Afs = (G, c, c, 7*, S) coincides with that of all the optimal submodular flows in the current J\f = (G,c,c,7, S). Consequently, (5.170) and (5.171) are valid for any optimal submodular flow p in the current J\f. Hence, the modifications of c and c in Step (4-1) do not change the set of all the optimal submodular flows in M and Step (4-2) maintains Property (1) given in the present theorem. Therefore, in Step (4-3) a maximal chain 0 = U^ C U{ C C C/|. = Wi of upper ideals of W is a chain of tight sets for dip restricted to Wl, i.e., d
(s = 0 , l , - - - , k l ) .

(5.173)

Since Property (1) in the present theorem is valid before Step (4-3) is executed, (5.173) holds for any optimal submodular flow ip in the current M. This validates the decomposition of (T>1, f%) into minors (T)1, f") \J\jU\_i (s = 1, 2, , ki) for each i G / in Step (4-3). This shows the validity of (1) and (2) at the next stage. Q.E.D. It follows from Theorem 5.14 that the algorithm terminates after at most \A\ + IV^| (IV| — 1) applications of Fundamental Cycle. When M = 0, the set of all the submodular flows in the current network is exactly the set of all the optimal submodular flows in the original network. An optimal submodular flow is obtained by finding a (feasible) submodular flow in the current network by use of the common-base algorithm described in Section 4.3 (also see Section 5.3). All the steps of Fundamental Cycle can be carried out in strongly polynomial time, provided that an oracle for exchange capacities is available. [For recent developments in submodular flow algorithms see [Iwata97], [Wallacher + Zimmermann99], [Fleischer + Iwata + McCormick02] and [Iwata + McCormick + Shigeno03] (also see [Fuji + IwataOO]). Also, see [Frank93, 96, 97, 98, 05] for theory and applications of submodular flows and their generalizations.] It should be noted that an exchange capacity can be computed in strongly polynomial time by applying the strongly polynomial algorithm [Grotschel + Lovasz + Schrijver88] for submodular-function minimization, where only an oracle for function evaluations is assumed, but the algorithm heavily relies on the ellipsoid method [Khachiyan79, 80] and is not a com-

188

III. NEOFLOWS

binatorial one. [See Section 14 in Chapter VI for recent developments in submodular function minimization.!

5.6. Matroid Optimization We show some specializations of the results obtained in the previous sections to matroids, which is a retrospective view of matroid optimization.

(a) Maximum independent matchings Let G = {V+ ,V~;A) be a bipartite graph with the left and right endvertex sets V+ and V~ and the arc set A. Also let M + = (V+,l+) and M~ = (V~,T~), respectively, be matroids on V+ and V~ with families 1+ C 2V+ and 1~ C 2V~ of independent sets. Denote M = (G = (V+,V-;A),M+,M-). An independent matching M C A in J\f is a matching in G such that <9 + M€X + ,

d-M<El-,

(5.174)

where d+M (d~M) is the set of end-vertices in V+ (V~) of arcs in M. (We assume that for each arc a £ A we have d+a £ V+ and d~a € V~.) The maximum independent matching problem is to find a maximum independent matching (i.e., an independent matching of maximum cardinality) in J\f. The maximum independent matching problem can naturally be reduced to a maximum independent flow problem as follows. Consider a network M = (G = (V+,V-;A),c,c,(2v+,p+),(2v+,p-)), where V+ (V~) is the set of entrances (exits), (2V ,p + ) ((2V ,p~)) is the submodular system on V+ (V~) with p+ (p~) being the rank function of matroid M + (M~), and c(a) = 0 (a € A), c(a) = +oo (a € A).

(5.175) (5.176)

We see that any integral independent flow ip in J\f is {0, l}-valued and that {a | a £ A, (p(a) = 1} is an independent matching in J\f. From Theorem 5.9 an integral maximum independent flow in J\f gives a maximum independent matching in jV and we have the following max-min theorem.

5.6. Matroid Optimization

189

Theorem 5.15 (The maximum-independent-matching minimum-coveringrank theorem) [Edm70], [Rado42], [Welsh70]: For network M = (G = (V+,V~; A), M+,lVr) we have max{|M| | M is an independent matching in Af} = mm{p+(U+) + p-(U-) | (U+,U-) is a cover of G}. (5.177) (Proof) The present theorem immediately follows from Theorem 5.9, where a cut U of finite capacity of AA corresponds to a cover (U+, U~) of G such that U+ = V+—U and U~ = V~dU; this gives a one-to-one correspondence between the set of cuts of finite capacity of J\f and that of covers of G, and the capacity of a cut U of finite capacity of A/" is equal to p+ (U+) + p~ (U~), the rank of the cover ([/+, U~). Q.E.D. The transformation of matroids by bipartite graphs Let G = (V+,V~;A) be a bipartite graph and M + = {V+ , J + ) be a matroid with a family X+ of independent sets. Define J - = {d~M | M: a matching in G, d+M € X + },

(5.178)

where d~M = {d~a \ a € M}. We can easily see that T~ is a family of independent sets of a matroid. From Theorem 5.15 the rank function p~ of the matroid M~ = (V~,T~) is given by

p-(U-) = mm{p+(X+) + \Y-\ \ (X+,Y~ U (V~ - U~)) is a cover of G} (5.179) for each U~ C V~. We call M~ the matroid induced from M + by the bipartite graph G. The matroid intersection problem When V~ is a copy of V+ and a bipartite graph G = {V+, V~; A) represents the natural bijection between V+ and V~, the maximum independent matching problem for the network M = (G = (V+, V~; A), M + , M~) becomes the problem of finding a maximum common independent set of the two matroid M + and M~, where V+ is identified with V~. This problem is called the matroid intersection problem. From Theorem 5.15 we have

III. NEOFLOWS

190

Theorem 5.16 (The matroid intersection theorem) [Edm70]: For two matroids M + = (E,l+) and M " = (E,l~) with T+ and T~ being families of independent sets, max{|/| | / e i + n r } = min{p+(X) + p-(E - X) | X C E},

(5.180)

where p+ (p~) is the rank function of M + (M~). Theorem 5.16 also follows from Theorem 4.9. The matroid intersection problem is a special case of the maximum independent matching problem in a natural way. Conversely, we can show that the maximum independent matching problem is reduced to a matroid intersection problem. Consider the maximum independent matching problem for a network N = (G = (V+, V~; A), M + , M~). From G define a bipartite graph G = (W+, W~; A), where W+ = {w+ | a € A}, A = {(w+,w-)

W~ = {w~ | a € A}, \aeA}.

(See Fig. 5.5.)

Figure 5.5: Graphs G and G. Moreover, define

(5.181) (5.182)

5.6. Matroid Optimization

X+ = {/+ I / + c W+,

191

W e V+: \{a\ae

5+v, w+ € 7+}| < 1,

{d+a | aE A, w+ € / + } e X+},

(5.183)

J " = {/- | I " C VF", Vt> e V~: \{a | a € (Tv, to" e / ~ } | < 1, {<9~a | a e A, w~ el~} e I ~ } ,

(5.184)

where <9+, d~, 5+ and 6~ are with respect to G. We can easily show and M~ = (W~,l~) are matroids with families that M + = (W+,l+) + 1 and X~ of independent sets. The matroid M (M ) is regarded as the one induced from M + (M~) by a bipartite graph, or as a composition of M + (M~) and the direct sum of rank-one uniform matroids on 5+v (v £ V+) (S~v (v £ V~))- The maximum independent matching problem for network M = (G = (V+, V~; A), M + , M~) is thus reduced to a maximum independent matching problem for the new network J\f = (G = (W+, W~; A), M , M ), which is a matroid intersection problem. The matroid union Let Mj = (Ei,Ii)

(i = 1, 2) be two matroids. Define j l v 2 = {i, u h I h e Xi, h e x 2 } .

(5.185)

Then Mi V 2 = (E1UE2, X1V2) is a matroid, which is the one induced from the direct sum M i © M2 of the two by the bipartite graph G = {V+, V~;A), where V+ = E\ © E2, V~ = E\ U E2 and the arc set A consists of the natural bijections between E\ C V+ and E\ C V~ and between Ei C V+ and E2 C V~ (see Fig. 5.6). Therefore, from (5.179) we have Theorem 5.17: The rank function piV2 of Mi V 2 is given by plv2{X) = mm{p1(YnE1) + p2{YnE2)

+ \X-Y\

\ Y C X}

(5.186)

for each X C Eil)E2. The matroid Mi V 2 is called the union (or sum) of Mj (i = 1,2). Note that if i?i = E2, we have /)iv2 = (pi + P2) , which is the rank function of the reduction of the sum (2E, pi + p2) of submodular systems (2E, p^ (i = 1,2) by vector 1 = (l(e) = 1 | e G E) (see (3.9)).

192

III. NEOFLOWS

Figure 5.6: Matroid union. A base of the union Miv2 of M i and M2 is given in the form of the union B\ U B2 of some bases f?j of matroids Mj (i = 1,2). When E\ = E2, for bases B{ (i = 1,2) of matroids M , B\ UB2 is a base of Miv2 if an-d only if B\ Pi (E2 — B2) is a maximum common independent set of M i and the dual M2* of M2. In this sense the problem of finding a base of the union of two matroids is equivalent to the matroid intersection problem and hence to the maximum independent matching problem. When we are given matroids Mj = (i?j,Zj) (i = 1,2, ) (k > 2), we can define the union MiV2v---vfc of Mj (i = 1,2, , fc) in the same way as in the case of k = 2. That is, an independent set of the union of Mj (i = 1,2, , k) is the union of independent sets /j € Zj (i = 1, 2, , A;). The rank function piv2v---vfc °f Miv2v---vfc is given by Piv2v-vk(X)

= mm{Pl(YnE1)

for each X C E1 U

+ --- + pk(YnEk)

+ \X-Y\

| Y C X} (5.187)

U Ek.

Theorem 5.18: There exist disjoint bases of Mj = (E,li) (i = 1, 2, one from each Mj, if and only if Pl(E)+---+pk(E)

= mm{Pl(X) + ---+pk(X) + \E-X\

, k),

\ X C E}. (5.188)

(Proof) The present theorem follows from the fact that the rank of the

5.6. Matroid Optimization

union of M, (i = 1,2,

193

is equal to the right-hand side of (5.188). Q.E.D.

The matroid partitioning For matroids M j = (E,li) with rank functions pi (i = 1,2, , the matroid partitioning problem is to find k disjoint subsets lj of E such that Ii U I2 U U 4 = E and / , € 2, (i = 1, 2, , k). We can easily see that k such subsets lj (i = 1, 2, , k) exist if and only if the union of M j (i = 1, 2, , k) is the free matroid on £J, i.e., from (5.187) \E\ = min{pipO +

+ pk(X) + \E - X\

\ X C £}.

(5.189)

In other words, Theorem 5.19 [Edm70]: There exists a base f?j of Mj = (E,Zi) for each i = 1, 2, , k such that the union of Bi (i = 1, 2, , k) is E if and only if VX C E: pi{X) +

pk(X) > \X\.

(5.190)

When matroids M j = (E,T{) (i = 1,2, ) are the same matroid M = (E, X) with the rank function p, E is partitioned into k independent sets I{ E I (i = 1, 2, , k) if and only if VX C E: kp{X) > \X\.

(5.191)

From (5.191) we have Corollary 5.20 [Edm65] (see [Tutte61], [Nash-Williams61] for graphs): The minimum number k for which E is partitioned into k disjoint independent sets of M = (E,l) is equal to max{\\X\/p(X)]

I 0 ^ X C E},

(5.192)

where we assume M does not have any selBoop, i.e., we assume p({e}) > 0 (e € E). Note that (5.191) is equivalent to that a uniform vector (l/k)l is an independent vector of the matroidal polymatroid P = (E, p) corresponding to M.

194

III.

NEOFLOWS

If the matroidal polymatroid P = (E,p) has a uniform base (l/k)l for positive integers k, I such that l\E\ = kp(E), then there exist k bases B{ (i = 1, 2, , k) of M such that each e E E is uniformly covered by Bi (i = 1, 2, , k) I times, i.e., for each e £ E \{i | i € {1, 2,

, k}, e £ £ i } | = Z,

(5.193)

due to the fact that B(p) + + B(p) (the sum of k copies) = B(fcp) with Z as the underlying totally ordered additive group (see Section 3.1.c). Such a family of bases Bi (i = 1, 2, , k) is called a complete family of bases of M and a matroid having a complete family of bases is called irreducible [Tomi75,76]. The concept of irreducible matroid was introduced by N. Tomizawa for the analysis of principal partition of a matroid (see Sections 7.2.b.l and 9.2). The same concept was also independently introduced by H. Narayanan [Narayanan74] and was called a molecule (also see [Narayanan + Vartak81] [and [Narayanan97]]). An algorithm for the maximum independent matching problem by using auxiliary graphs is given by Tomizawa and Iri [Tomi+Iri74]. An algorithm for the matroid intersection problem is given by Edmonds [Edm79]. For other related algorithms see [Edm+Fulkerson65] for matroid partitionings and [Bruno+Weinberg71] for matroid unions.

(b) Optimal independent assignments Consider a bipartite graph G = (V+,V~;A), matroids M + = (V+,I+) and M~ = (V~,I~) and a weight function w: A —> R. Denote such a network by M = (G = (V+, V~; A),M+,M~,w). For a positive integer k, a k-independent matching M in A/" is an independent matching of cardinality k in A/"0 = (G = (V+ ,V~;A), M + , M~). An optimal k-independent assignment in M is a ^-independent matching M having the minimum weight w(M) = J2eeEw(e) among all the ^-independent matchings in J\f. The independent assignment problem is to find an optimal fc-independent assignment in M for a given k. The independent assignment problem is a special case of a neoflow problem, especially of the independent flow problem in a natural way. The independent assignment problem is equivalent to the weighted matroid intersection problem, which is to find a minimum-weight common independent set, of two matroids, having cardinality k for a given positive integer k.

5.6. Matroid Optimization

195

For an independent matching M define the auxiliary network MM = = {V*,AM),WM) as follows. GM is the graph with vertex set V* = F + u r u {s+, s"} and arc set AM = SM U A ^ U AM U M U ^ M U SM, where (GM

S1^

=

A^

= {(u, « ) | « € cl+(9+M) - a+M, u G C+(a+M|w) - {u}}, (5.195)

AM M

= A-M, = {a | a G M}

AM

= {(v,u) | v € cl~(9~M) - d~M, u G C~(9~M|u) - {v}}, (5.198)

1

S^

=

{(s+, v) \veV+

- cl+(<9+M)} U {(v, s+) | u G a + M } ,

(a: a reorientation of a),

{(v,s~)\v&V~-d~(d~M)}U{(s~,v)\ved~M}.

(5.194) (5.196) (5.197) (5.199)

Here, cl + and cl~ are, respectively, the closure functions of M + and M ~ , C+(d+M\v) for v € cl+(d+M)-d+M is the fundamental circuit associated + + with d M G X and v which is the unique circuit of M + contained in d+MU{v}, and C~(d~M\v) is similarly defined for M ~ . Also, wM- AM — R is the length function defined by

{

w{a) —w(a) 0

(a € AM) (a<EM, a(e M): a reorientation of a) (a(ESMUA+IuAMUSM).

(5.200)

A primal-dual algorithm for the independent assignment problem [Iri+Tomi76] Step 1: Put M <- 0. Step 2: While \M\ < k, do the following. (2-1) Find a shortest path P, relative to the length function WM, from s+ to s~ in the auxiliary network MM having the minimum number of arcs. If there exists no directed path from s+ to s~ in MM, then stop (there is no fc-independent matching). (2-2) Put M <- M A A(P) (A: the symmetric difference), where A(P) = {a | a G A, a lies on P} U{a | a G .M, a reorientation of a lies on P}. (5.201)

196

III.

NEOFLOWS

(End) If we define in Step (2-1) a potential p: V* —> R in such a way that for each v € V* p(v) is equal to the length of a shortest path from s+ to v in MM, then using the potential p we can replace WM in the next execution of Step (2-1) by WM,P defined by WM,p(a) = WM(O-) +p(d+a)

—p(d~a).

(5.202)

Here, we disregard those vertices which are not reachable from s + in MM, since vertices which are not reachable from s+ will not become reachable from s + in MM for new M. We can show that thus denned WM,P is nonnegative for those arcs reachable from s+, so that we can reduce the complexity of finding a shortest path in MM (see [Iri + Tomi76]). This is an adaptation of the technique for the classical min-cost flows developed by Tomizawa [Tomi71] and also independently by Edmonds and Karp [Edm + Karp72]. It should be noted that arcs entering s+ or leaving s~ in MM play no role in the primal-dual algorithm. These arcs are for the primal algorithm given as follows.

A primal algorithm for the independent assignment problem [Fuji 77a] Step 1: Find a /c-independent matching M in M. Step 2: While there exists a negative cycle, relative to the length function 4, in the auxiliary network MM, do the following. (2-1) Find a negative cycle Q in MM having the minimum number of arcs. (2-2) Put M <- M A A(Q), where A(Q) is denned by (5.201) with P replaced by Q. (End) For other related algorithms, see [Edm79], [Lawler75], [Fuji77b], [Iri78], [Frank81a]. The problem of finding a minimum weight directed spanning tree in a graph is an example of the independent assignment problem or the weighted matroid intersection problem. Consider a graph G = (V, A) and a weight function w: A —> R. Let M i be the graphic matroid M(G) represented by G

5.6. Matroid Optimization

197

and M2 be the direct sum of the rank-one uniform matroids M~(w) on S~v (v € V), i.e., M 2 = ®v€VM~(v). We see that B C A with \B\ = \V\ - 1 is a common independent set of Mi and M2 if and only if B forms a directed spanning tree of G. Hence the problem of finding a minimum weight directed spanning tree is a weighted matroid intersection problem. If we want to find a minimum weight spanning tree with a fixed root VQ € V, then replace M~(i>o) by the trivial matroid (rank-zero matroid) on 5~VQ. For other applications, see [Iri83], [Iri + Fuji81], [Murota87] and [Recski89] [(also see [Narayanan97] and [MurotaOOa])]. Finally, it should be noted that the problem of finding a common independent set of maximum cardinality for three matroids is NP-hard. For, consider a graph G = (V, A), and let Mi be the 1-elongation of the graphic matroid M(G) and M2 be the direct sum of the rank-one uniform matroids M~(u) on 8~v (v € V), where the 1-elongation of M(G) is the union of M(G) and the rank-one uniform matroid on A (see Section 3.1.d). Also, let M3 be the direct sum of rank-one uniform matroids M + (u) on 5+v {v € V). We can easily see that the maximum cardinality of common independent sets of Mj (i = 1,2,3) is equal to \V\ if and only if there exists a directed Hamiltonian cycle in G, and that a common independent set of Mj (i = 1, 2, 3) of cardinality |V| forms a directed Hamiltonian cycle in G.

This Page is Intentionally Left Blank

199

Chapter IV. Submodular Analysis Submodular (or supermodular) functions on distributive lattices share similar structures with convex (or concave) functions on convex sets. In this chapter we develop a theory of submodular and supermodular functions from the point of view of the duality in convex analysis ([Fuji84f, 84g, 84h]).

6. Submodular Functions and Convexity We define the convex (concave) conjugate function of a submodular (supermodular) function and show a Fenchel-type duality theorem for submodular and supermodular functions. We also define the subgradients and subdifferentials of a submodular function and examine the relationship among these concepts and the polyhedra such as the submodular and supermodular polyhedra and the base polyhedron associated with the submodular function. The reason for the analogy between a submodular function and a convex function is nicely explained by the Lovasz extension of a submodular function.

6.1. Conjugate Functions and a Fenchel-Type Min-Max Theorem for Submodular and Supermodular Functions (a) Conjugate functions Let / : D ^ R b e a submodular function and g: V R be a supermodular E function on a distributive lattice V C 2 with 0, E £ V. Here, we do not necessarily assume that /(0) = g(9) = 0. Define the function /*: RE -> R by f*(x) = max{x(X) - f(X) \ X € V}

(x € ~RE)

(6.1)

and also, in a dual form, the function g*: R —> R by g*(x) = min{x(X) - g(X) \ X e V}

(x € RE).

(6.2)

200

IV. SUBMODULAR

ANALYSIS

By the definition / * is a convex function and g* is a concave function. We call / * the convex conjugate function of the submodular function / and g* a concave conjugate function of the supermodular function g. It may be noted that the terms x(X) appearing in (6.1) and (6.2) can be regarded as the inner product of x and xx, the characteristic vector of X. Notice the analogy between the definition (6.1) and that of a convex conjugate function of an ordinary convex function (see (1-35)) ([Rockafellar70], [Stoer+Witzgall70]). Theorem 6.1: For a submodular function f: T> —> R and a supermodular function g: T> —> R we have for any X G T> f(X) g(X)

= max{x(X) - /*(x) | x € KE}, E

= min{x(X) - g*(x) | x € H }.

(6.3) (6.4)

Moreover, for any X € 2E — V x(X) — f*(x) (or x(X) — g* (x)) as a function of x € R S can be made arbitrarily large (or small). (Proof) Without loss of generality we assume that /(0) = x(X)-f(x) (6.5) for any X € T> and x € R . We show that for any X € T> there exists a vector x € R such that (6.5) holds with equality, from which (6.3) follows. From Lemma 3.2, for any I e D there exists a subbase x € P(/) such that x(X) = f(X). Since x € P(/), we have from (6.1) f(x) = x(X)-f(X)(=0).

(6.6)

This implies (6.3). Relation (6.4) is the same as (6.3) by considering the dual order on R. Moreover, it follows from Lemma 3.2 that for any X € 2E — T> we can make x(X) arbitrarily large (or small) subject to x € B(/) (or x € B(g)). Since f*(x) = 0 for x € B(/) and g*(x) = 0 for x € B(g), this completes the proof. Q.E.D. We see from Theorem 6.1 that the correspondence between a submodular (or supermodular) function / (or g) and its convex (or concave) conjugate function / * (or g*) is one to one.

6.1. Conjugate Functions and A Fenchel-Type Theorem

201

The convex conjugate function /* is closely related to the vector rank function if of the submodular system (P, / ) when /(0) = 0. Recall that for each x £ HE Tf(x) = min{f(X) + x(E-X)\XeV}

(6.7)

(see (3.9) or (3.20)). Lemma 6.2: For a submodular system (T>, / ) on E and a vector x € R we have

f*(x) = x(E)-rf(x). (Proof) Immediate from (6.1) and (6.7).

(6.8) Q.E.D.

Note that since ij is submodular on the vector lattice HE (see [Welsh76] when V = 2E), the convex function /* is supermodular on ~RE, i.e., for any x, y € RE f*(x) + f*(y) < f*{x \Jy) + f*(x A y),

(6.9)

where (xVy)(e) = max(x(e), y(e)), (xAy)(e) = min(x(e), y(e)) (e € E). (See [Topkis78] for submodular functions on general lattices.) Here, it may also be noted that vector lattice R^ is a distributive lattice. (b) A Fenchel-type min-max theorem

We show a Fenchel-type min-max theorem for submodular and supermodular functions, which relates the difference between a submodular function / and a supermodular function g to that between the concave conjugate function g* and the convex conjugate function /*. (For Fenchel's duality theorem for ordinary convex and concave functions, see [Rockafellar70] and [Stoer + Witzgall70].) Theorem 6.3 (A Fenchel-type min-max theorem) [Fuji84f]: For a submodular function f: T>\ R and a supermodular function g: T>2 R on distributive lattices Z>i, V2 C 2E with 0, E £ Vx n T>2 we have min{f(X) - g{X) \ X € V1 n V2) = max{g*(x) - /*(x) | x € KE}.

(6.10)

202

IV. SUBMODULAR

ANALYSIS

Moreover, if f and g are integer-valued, the maximum on the right-hand side of (6.10) can be attained by an integral vector x. (Proof) Without loss of generality we assume that /(0) =ff(0)= 0. Hence (£>i, / ) is a submodular system and (T>2,g) is a supermodular system, both on E. The min-max relation (6.10) is equivalent to mm{f(X) + g*{E - X) \ X G V1 n V2} = max{g*(x) + g{E) - f*(x) \ x G RE},

(6.11)

where recall that g& is the dual submodular function of g. From (6.2), (6.7) and Lemma 6.2, f*(x) = x(E)-Tf(x), (6.12) g*(x)+g(E) =

mm{x(X)+g*(E-X)\XEV2}

= rg#(x).

(6.13)

Substituting (6.12) and (6.13) into (6.11) yields min{/(X) +g#(E-X) \X eViD V2} = max{r/(x) + rg#(x) - x{E) \ x € HE},

(6.14)

which is equivalent to (6.10). For any x € HE let y and z be bases of the reductions of (Pi, / ) and (T>2,g^) by x, respectively, i.e., yeP(/),

y<x,

y(E) = rf(x),

(6.15)

z€P(ff#),

z<x,

z(E) = rg#(x).

(6.16)

Also define w = y A z (= (min{y(e), z(e)} \ e G E)).

(6.17)

Then, wG P ( / ) n P ( f f # ) .

(6.18)

Because of (6.15)~(6.18), if(x) +

rg#(x)-x(E)

= y{E) + z(E) - x(E) < w{E) +w{E) -w{E) = if(w) + rg#(w) - w(E) (= w{E)).

(6.19)

6.1. Conjugate Functions and A Fenchel-Type Theorem

203

It follows from (6.18) and (6.19) that (6.14) (or (6.11)) is also equivalent to

min{f(X) + g*{E - X) | X € Vx n V2} = max{x(£) | x € P(/) D P(g*)}-

(6.20)

The min-max relation (6.20) is exactly the intersection theorem (Theorem 4.9), so that (6.10) holds. The integrality part of the present theorem follows from the counterpart of the intersection theorem. Q.E.D. Note that if x € R is a maximizer of the right-hand side of (6.10), then w given by (6.15)^(6.17) is a maximizer of the right-hand side of (6.20). If / and g are integer-valued functions and x is an integral vector, then such a vector w can also be integral. Conversely, any maximizer x of the right-hand side of (6.20) is a maximizer of the right-hand side of (6.10). The above proof shows the equivalence between the Fenchel-type minmax theorem and the intersection theorem. Combining this with the results in Sections 4.1 and 4.2, we see that the three theorems, the intersection theorem (Theorem 4.9), the discrete separation theorem (Theorem 4.12) and the Fenchel-type min-max theorem (Theorem 6.3), are equivalent. The Fenchel-type min-max theorem motivates further investigation of submodular and supermodular functions from the point of view of the duality theory in convex analysis ([Rockafellar70], [Stoer + Witzgall70]). 6.2. Subgradients of Submodular Functions (a) Subgradients and subdifferentials Consider a submodular function / : V —> R on a distributive lattice V C 2E with 0, E E V. For a vector x € KE and a set X € V, if x(Y)-x(X)
(6.21)

holds for each Y € V, then we call x a subgradient of / at X. We denote by df(X) the set of all the subgradients of / at X and call df(X) the subdifferential of / at X. Previously we employed symbol d as the boundary operator for flows in networks. Here we use the same symbol for subdifferentials, following the convention in convex analysis, because there seems to be no possibility of confusion.

204

IV. SUBMODULAR ANALYSIS

Figure 6.1: Subdifferentials. Figure 6.1 shows a two-dimensional example of subdifferentials of / : V —> R w i t h D = 2^'2>. In general, HE is divided into \T>\ unbounded polyhedra df(X) (X E V) and for distinct X, Y G T> the subdifferentials df(X) and df(Y) may have common faces but not common interior points. [Note that the convex conjugate function /* of / is a convex function on HE such that /* is affine on df(X) for each X € V and that df(X) (X € V) are g-polymatroids as will be seen below. That is, /* is a special M^-convex function, which will be considered in Section 17.] It may be noted that a subgradient of any set function can be denned by (6.21). Some of the arguments in the following are valid for any set function. Lemma 6.4: For a submodular function f: V have x € df(X) if and only if x € R, satisfies x{Y) - x(X) < f(Y) -

R and a set X G V, we f(X)

(6.22)

for each Y € [0, X]v U [X, E]v, where [
{Y\YeV,YCX},

(6.23)

6.2. Subgradients of Submodular Functions

205

[X, E]v = {Y | Y e V, X C Y}.

(6.24)

(Proof) It is sufficient to show the "if" part only. Suppose that (6.22) holds for each Y G [0, X]v U [X, E]v. Then for any Z e P , x(X U Z) - x-(X) < f(X U Z) - / ( X ) ,

(6.25)

x(inz)-x(i)
(6.26)

From (6.25), (6.26) and the submodularity of / , x{Z)-x{X)

= x(X U Z) - x(X) + x(X C\ Z) - x(X)

< f(xuz) + f(xnz)-2f(x) <

f(Z)-f(X). Q.E.D.

Lemma 6.5: For a submodular function f-.V^H, have

and a set X € T> we

df(X) = dfx(X)xdfxm, x

x

(6.27)

E X

where df {X) C R , dfx(Q) C R, ~ , x denotes the direct product, x and f and fx are, respectively, the submodular functions on [0, X]-p and [X, E]v/X = {Y -X \Y e[X, E}v} defined by fx(Y)

= f(Y) (Y(E[(D,X]V),

fx(Y) = f(XUY)-f(X)

(Ye[X,E]v/X).

(6.28) (6.29)

(Proof) From Lemma 6.4, we have x 6 df(X) if and only if x(Y)-x(X)

<

f(Y)-f(X) x

= f (Y)-fx(X) x(Z)-x(0)

=

x(ZUX)-x(X)

<

f(ZUX)-f(X)

= fx(Z) - fx()

(Ye[
(Z£[X,E]V/X).

(6.30)

(6.31)

(6.30) means xx (= (x(e) \ e e X)) e dfx(X) and (6.31) means xE~x (= (x(e) \eeE-X)) e dfx(Q). We thus have (6.27). Q.E.D.

206

IV. SUBMODULAR

ANALYSIS

Lemma 6.6 (cf. [Rockafellar70, Theorem 23.5]): For a (submodular) function f: T> —» R, a vector x G R and a set X G T>, the following three are equivalent:

(i) (ii) (iii)

x G df{X), mm{f(Y) + x{E-Y) \ Y eV} = f(X)+ x(E - X), f(X) + f*(x) = x(X).

(6.32) (6.33) (6.34)

(Proof) We can easily see that each of the above three is equivalent to

min{/(Y) -x{Y) \ Y G V} = f(X) - x(X).

Q.E.D.

Lemma 6.6 holds for any set function. Lemma 6.7: For a submodular function f: V

R with 0, E € V and

/(0) = o, (a)

5/(0) = P(/),

(6.35)

(b) (c)

df(E) = P(f#), 0/(X)nB(/)^0

(6.36) (6.37)

(XEV).

(Proof) (a) and (b) immediately follow from the definition of subdifferential. We prove (c). For any X 6 T> there is a base x of the submodular system (£>,/) such that x{X) = f(X) (see Lemma 3.2). Since x G B(/) and x(X) = f(X), it easily follows from (6.21) that x e df(X). Hence, df(X)D

B(/) + 0.

Q.E.D.

When /(0) = 0, (6.27) implies that the subdifferential df{X) is the direct product of the supermodular polyhedron dfx(X) and the submodular polyhedron dfx{$), due to Lemma 6.7. The following theorem is a submodular analogue of [Rockafellar70, Theorem 23.8] in convex analysis. Theorem 6.8: Let f\:T>\ —> R and j^'-T^i ^ R be submodular functions, where V\ and T>2 are distributive lattices with 0, E G T>\ fl T>i- Then, for each X G T>\ n T>2 we have d(fi + h){X) = 8h{X) + df2(X).

(6.38)

Also, for each A > 0 and X G £>i, d(Xfi)(X)

= Xdh(X).

(6.39)

6.2. Subgradients of Submodular Functions

207

Moreover, for A = 0 and X € V\, 0(0 fi)(X)

= {x | x € HE, VY € Vi: x(Y) - x(X) < 0} =

where 0+dfi(X) (see (1.22)).

0+dfi(X),

(6.40)

is the characteristic cone (or recession cone) of dfi(X)

(Proof) From Lemma 6.5, for each % = 1,2 and X £ T>i dfi(X) is the direct product of dfiX(X) and dfiX(9). It follows from (3.32) (and its dual counterpart for supermodular functions) and from Lemma 6.7 that for each X £ T>\ n T>2 we have dtfi + h){X)

= = = = =

d(fl + f2)x(X)xd(f1 + f2)x(H}) x x d(h + f2 )(X) x d(flx + f2X)(9) (dfrx(X) + dhx(X)) x (9/ lx (0) + 0/2*(0)) ( d / i x p 0 x 0/ix(0)) + (9/ 2 x (X) x 0/2*(0)) (6.41) df\{X) + df2{X).

Relations (6.39) and (6.40) are immediate from the definition of subdifferential. Q.E.D. Note that we have used the intersection theorem through (3.32) to prove Theorem 6.8. For a convex conjugate function /* of a submodular function / : T> —> R and a vector x £ R B , define the set d2f*(x) of subsets of E as follows: X € 0a/*(x)

(6.42)

y(X)-x(X)
(6.43)

if and only if X C E and

for each y € R, . We call d2f*(x) the binary subdifferential of /* at x. Note that from (6.43) and Theorem 6.1 each X £ d2f*(x) belongs to V. Theorem 6.9 (cf. [Rockafellar70, Theorem 23.5]): Consider a submodular function f:T> — R. For any vector x E HE and any set X £ V the following two are equivalent.

(i) x € 0/pO,

(6.44)

208

IV. SUBMODULAR

(ii) X €

ANALYSIS

d2f*(x).

(6.45)

Moreover, the binary subdifferential <92/*(x) is a sublattice ofD. (Proof) From Theorem 6.1, Lemma 6.6 and the definition (6.43) of binary subdifferential of /*, we see that both (i) and (ii) are equivalent to f(X) + f*(x) = x(X).

(6.46)

Moreover, (6.45) or equivalently (6.46) implies x(X)

- f(X) = m a x { x ( y ) - f(Y) \YeV}.

(6.47)

Therefore, <92/* (x) is the set of the maximizers of the supermodular funcQ.E.D. tion x — f on T> and hence is a sublattice of T>. For two submodular functions ff.Vi —> R (i = 1,2) define the convolution / i * o /2*:R —> R of the convex conjugate functions f\* (i = 1,2) by (/l* o /2*)(x) = min{/i*(xi) + / 2 *(x 2 ) | Xi + x2 = x} (6.48) for each x £ HE. Theorem 6.10 (cf. [Rockafellar70, Theorem 16.4]): For two submodular functions ff. V{ -> R (i = 1,2) with 0 € P i fi P 2 we have / i * o / 2 * = (/ 1 + / 2 )*.

(6.49)

(Proof) For any x E~RE, (/i + / 2 )*(x) = max{x(X) - h(X) - f2(X) \ X € V1 n P 2 } < m i n { m a x { x 1 ( X ) - / 1 ( X ) + x 2 ( y ) - / 2 ( y ) | X € Pi, y € P 2 } Xi + X2 = x} = min{/i*(xi) + /2*(x2) | xi + x2 = x} = (/i*o/ 2 *)(x).

(6.50)

On the other hand, let x be any vector in R^. There exists X € Pi fl P 2 such that x € <9(/i + / 2 )(X). Furthermore, from Theorem 6.8 there exist

6.2. Subgradients of Submodular Functions

209

x

i € dfi(X) and x2 € df2(X) such that x\ + x2 = x. Consequently, from Lemma 6.6 (/i + /2)*(x)

= x{X) - h{X) - f2{X) =

x1(X)-f1(X)+x2(X)-h(X)

= fi*(x1) + f2*(x2) >

(/i*o/2*)(x).

We thus have (6.49).

Q.E.D.

(b) Structures of subdifferentials Suppose that V C 2E is a simple distributive lattice, i.e., V = 2V for a poset V = (E, ;^). Consider a submodular function / : V —> R. We first give a characterization of the extreme points of the subdifferential df(X) for X € V. Theorem 6.11: For each X E T>, x E R s is an extreme point ofdf(X) if and only if there exists a maximal chain C: 0 = 5 o c 5 i C

n

=E

(6.51)

ofV, including X in it, such that x(5i - 5 i _ i ) = / ( 5 i ) - / ( 5 i _ i )

(i = l , 2 , - - - , n ) .

(6.52)

(Proof) We assume, without loss of generality, that /(0) = 0. From Lemma 6.7 we have <9/(0) = P(/). Note that P(/) and B(/) have the same set of extreme points. Therefore, the present theorem for X = 0 follows from Theorem 3.22 and, similarly, the present theorem for X = E follows from Theorem 3.22, since df(E) = P ( / # ) due to Lemma 6.7 and P ( / # ) and B(/#) (= B(/)) have the same set of extreme points. Furthermore, for any X EVwe have df(X) = dfx(X) x <9/x(0) due to Lemma 6.5. Hence the extreme points of df(X) are given by the direct product of extreme points of dfx(X) and those of dfx(®)- Since X is the unique maximal element of the domain [0, X]x> of fx and 0 is the unique minimal element of the domain [X, E]v/X of fx, the present theorem for X E V with l c l c £ Q.E.D. follows from that for X = E and X = 0 for / with domain V.

210

IV. SUBMODULAR ANALYSIS

The characteristic cone (or the recession cone) of a subdifferential df(X) (X E V) is given by

Cf(X) = {x x G RE, Vy G V: x(Y) - x{X) < 0} = {x x G RE, Vy G [0, X]v U [X, E}v: x(Y) - x(X) < 0}. (6.53) Note that Cf(X) depends only on V and X. We next give a characterization of extreme rays of Cf(X). Let G(V) = {E,B*(V)) be the directed graph with the vertex set E and the arc set B*(V) which represents the Hasse diagram of the poset V = (E, ^ ) , i.e., (e, e') € B*(P) if and only if e covers e' in V. Denote by E+ and E~, respectively, the set of all the maximal elements of V and the set of all the minimal elements of V. Note that E+ P\E~ may be nonempty. For each I e P denote by A~ (X) the set of all the arcs entering X in G(V) Define vectors (p+ (p+ E E+), rjp- (p~ E E~) and (a (a E B*(V)) in KE by

W ) = {"o1 tePE-{P+})

{P+eE+l

(jrG }

^'

VW = {I (ee£V» r Ca(e)

=

1

-

(6 55)

-

(e = e>)

<^ - 1 (e = e")

[ 0

(6 54)

(a = (e', e") G B*(7>)).

(6.56)

(ee£-{e',e"})

Also define for each X E V ER(X) = { V | p + € E+ - X} U {rip- \ p~ E E~ H X} U{Ca\aEB*(V)-A-(X)}.

(6.57)

Now, we are ready to show a theorem characterizing extreme rays of Cf(X) (X G V). Theorem 6.12: For each X G T>, the set of all the extreme rays of the characteristic cone Cf(X) of the subdifferential df(X) is given by ER(X) in (6.57). (Proof) We can easily see from (6.53) that ER(X) C Cf(X).

(6.58)

6.3. The Lovasz Extensions of Submodular Functions

211

Since no vector in ER(X) can be expressed as a nonnegative linear combination of the other vectors in ER(X), it suffices to prove that every vector in Cf(X) can be expressed as a nonnegative linear combination of vectors in ER(X). Let v be an arbitrary vector in Cf(X). From (6.53), v(X-Y)>0 (X^YEV),

(6.59)

v(Y-X)<0

(6.60)

(XCYGV).

Suppose that each arc of B*(V) — A~(X) has the infinite upper capacity and the zero lower capacity and that each arc of A~ (X) has the zero upper and lower capacities. Then it easily follows from (6.59), (6.60) and the feasibility theorem for network flows (Theorem 1.3) [Hoffman60] ([Ford + Fulkerson62]) that there exist a nonnegative flow
6.3. The Lovasz Extensions of Submodular Functions Consider a submodular function / : V —> R on a simple distributive lattice V = 2V with V = (E, ^ ) . We assume /(0) = 0. Define the convex function / : HE - ^ R U {+00} by /(c) = max{(c,x) | x e P(/)}

(6.62)

212

IV. SUBMODULAR

ANALYSIS

for each c G ~RE, where (c,x) = Ytc(e)x(e).

(6.63)

e£E

Here, / is called the support function of P ( / ) and is a positively homogeneous function, i.e., /(Ac) = A/(c) for each A > 0 and c € HE ([Rockafellar70], [Stoer + Witzgall70]). We see from Corollary 3.14 that /(c) < +oo if and only if c: E —> R is a nonnegative monotone nonincreasing function from V = [E,<) to (R, <). Therefore, for any c € RE such that /(c) < +oo there uniquely exist a chain A1cA2C---cAk of nonempty A\ G V (i = 1, 2, , k) such that

(6.64)

, k) and positive numbers Aj G R (i = k

c = X>**

(6-65)

where A; > 0, x^j £ R £ is the characteristic vector of Ai C E (i = 1,2, and if A; = 0 (i.e., c = 0), the empty sum is denned to be zero vector 0 in HE. Moreover, we have k

f{c) = YJhf{Al)

(6.66)

since the value of /(c) denned by (6.62) can be obtained by the greedy algorithm (see Section 3.2.b) and a maximizer x of (6.62) satisfies x(Ai-Ai-1)

= f{Ai)-f{Ai-1)

(i = l,2,---,k)

(6.67)

with AQ = 0. If the right-hand side of (6.66) is the empty sum, it is denned to be zero. Formula (6.66) was introduced by L. Lovasz [Lovasz83] for V = 2E. The construction of / through (6.64)~(6.66) can be applied to any function / on T> with /(0) = 0 and / is an extension of / . We call such an extension / the Lovasz extension of / [which is sometimes called the Choquet integral (see [Choquet55])]. Theorem 6.13 [Lovasz83]: A function f:V^~R only if the Lovasz extension f of f is convex.

is submodular if and

6.3. The Lovasz Extensions of Submodular Functions

213

(Proof) If / is a submodular function, then its extension / is given by (6.62) and hence is a convex function. Conversely, suppose that the extension / of / is a convex function. By definition, for any X, Y € T> f(xx + XY)

= fixxuY + XXC\Y) = f(XUY) + f(XnY).

(6.68)

Since / is a positively homogeneous convex function, we also have fiXx + XY) < f(xx) + RXY) = f(X) + f(Y). From (6.68) and (6.69) / is a submodular function on T>.

(6.69) Q.E.D.

Theorem 6.13 shows the close relationship between the submodularity and the convexity. The results in Sections 6.1 and 6.2 can be viewed from the theory of convex functions through this theorem. However, the integrality result in Theorem 6.3 does not follow directly from the ordinary convex analysis; it is truly a combinatorial deep result. Define P(£>) = the convex hull of vectors

XA (AGT>).

Lemma 6.14 [Lovasz83]: For a submodular function f:V^Hwe min{f(X) | X € V} = min{/(c) | c € P(2?)}.

(6.70) have (6.71)

(Proof) Immediate from Theorem 6.13 and (6.66), the positive homogeneity of / . Q.E.D. Lemma 6.15: For any c £ P(T>) there uniquely exists a nonempty chain Bi C B2 C

C Bp

(6.72)

ofT> such that c is expressed as a convex combination

c = it^XBl with in > 0 (i = 1,2,

,p) and Zf=1 Hi = 1.

(6.73)

214

IV. SUBMODULAR ANALYSIS

(Proof) Any c G ?(£*) is a nonnegative monotone nonincreasing function from V = (E, ^ ) to (R, < ) . Therefore, there uniquely exists a chain (6.64) of nonempty Ai G V (i = 1, 2, , k) and Aj > 0 (i = 1, 2, , k) such that (6.65) holds. Since c G P(V), we have k

( 6 - 74 )

]>>
If J2i=i ^i = I? then (6.65) is the desired unique expression. Otherwise put Ao = 1 — Yli=i ^i a n ( i AQ = $. This yields a desired unique expression k

c = J2^XA2

(6.75)

i=0

with Xi > 0 (i = 0,1,

, A;) and YA=O XI =1-

Q.E.D.

Now, let / : V —> R be a submodular function and define / : HE —> R U {+00} by /(c)

-\+oo

(ceRB-P(P)),

(6 76)

-

where / is the Lovasz extension of / . Call / the truncated Lovdsz extension of/. Theorem 6.16: For each A G V we have df(A) = df(XA),

(6-77)

where df(xA) denotes the subdifferential of the convex function f at XA in an ordinary sense of convex analysis (see (1.34)). (Proof) By definition, we have x £ df(xA) if and only if Vc G KE: (c - XA, x) < f(c) - f(XA).

(6-78)

Since /(XA) = f(A), (6.78) is rewritten as f(A)-x(A)

< min{/(c)-(c,x-)

c G RE}

= min{/(c)-(c,x)

cGP(P)}

=

min{/(X) - x{X) \ X eV},

(6.79)

6.3. The Lovasz Extensions of Submodular Functions

215

where the last equality follows from Lemma 6.14 with / replaced by / — x. (6.79) is equivalent to x € df{A). Q.E.D. T h e o r e m 6.17: Let c be an arbitrary vector in P(T>) and suppose that c is expressed as (6.73) with (6.72). Then, we have df(c) = f]{df(Bl)

\i = l,2,-..,p}.

(6.80)

(Proof) We have x € df(c) if and only if V b G P(V): (b-c,x) < f(b) - /(c).

(6.81)

From (6.72) and (6.73), (6.81) is rewritten as ^tHifiBj-xiBi))

< mm{f(b)-(b,x)\beP(V)}

i=l

=

min{f(X) - x(X) \ X eV}

(6.82)

due to Lemma 6.14. Furthermore, since YH=I fa = 1 a n ( i fa > 0 (i = 1,2, ,p), (6.82) is equivalent to f{Bi) - x(Bi)

= mm{f(X)

- x(X) \ X e V } (i = l , 2 , - - - , p )

(6.83)

or

xef]{df(Bi)\i

= l,2,---,p}.

(6.84) Q.E.D.

For any maximal chain C: 0 = SQ C S\ C C Sn = E of V, denote by P(C) the n-simplex with vertices xst {i = 0,1, , n). Lemma 6.18: The collection ofP{C)'s for all the maximal chains CofV forms a simplicial subdivision of P(T>). Moreover, for two maximal chains C Sln = E (i = 1, 2) the n-simplices P(Cl) (i = 1, 2) Cl: 0 = S'o C S\ C have a common facet if and only if for some k with 1 < k < n — I we have S} = S] (0<j
j^k).

(6.85)

(Proof) The first half of this lemma follows from the uniqueness property of Lemma 6.15. Moreover, any facet of the n-simplex P{Cl) corresponds to

216

IV. SUBMODULAR ANALYSIS

a subchain of Cl with length n — 1. From this follows the second half of the lemma. Q.E.D. Lemma 6.19: For any maximal chain C: 0 = So C Si C C Sn = E of V and any interior point c of P(C), / has a unique subgradient x at c given by x{Si-Si-1)

= f(Si)-f(Si-1)

(i = l,2,---,n).

(6.86)

(Proof) From Theorem 6.17 vector x given by (6.86) is the unique subgradient of / at c. Q.E.D. Note that the subgradient x of / given by (6.86) is an extreme point of the base polyhedron B(/). We can see that for a submodular function / : V —> R, the convex conjugate function /* and the truncated Lovasz extension / of / are the convex conjugate functions of each other in an ordinary sense of convex analysis (see (1.35)), since f*(x)

= max{i(I)-/(I)|X6D} = max{(c, x) - f(x) | x € R B } .

(6.87)

Consequently, the Fenchel-type min-max theorem for submodular and supermodular functions (Theorem 6.3), except for the integrality property, follows from Fenchel's duality theorem for ordinary convex and concave functions. Finally, it should be noted that the extension of the rank function of a polypseudomatroid can be denned by generalizing the Lovasz extension of a submodular function, due to the greediness property of polypseudomatroids (see [Qi88]).

7. Submodular Programs In this section we consider optimization problems with objective functions and constraints described by submodular functions, which we call submodular programs ([Fuji84f]). 7.1. Submodular Programs — Unconstrained Optimization

7.1. Unconstrained Submodular Programs

217

Let / : T> —> R be a submodular function on a simple distributive lattice T> C 2E with /(0) = 0. We consider the problem of minimizing the submodular function f:T> —> R without any constraints. It should, however, be noted that the underlying distributive lattice V itself may be regarded as a constrained feasible region. [We treat submodular function minimization in more detail in Chapter VI.]

(a) Minimizing submodular functions From the definition of subdifferential of a submodular function we have the following trivial but fundamental lemma. Lemma 7.1: A set A € V is a minimizer of f:T> —> R if and only if

o e df(A),

(7.1)

where 0 is the zero vector in HE. From Lemmas 6.4 and 7.1 we get the following theorem which means that some "local" or "partial" minimality implies "global" minimality. Theorem 7.2: A set A € T> is a minimizer of f:T> —> R if and only if A € V minimizes f restricted to the sublattice [0, A]v U [A, E]-p. Grotschel, Lovasz and Schrijver [Grotschel + Lovasz + Schrijver88] have devised a strongly polynomial algorithm for minimizing a submodular function / : T> —> R which requires time polynomial in |_E|. Their algorithm heavily depends on the so-called ellipsoid method [Khachiyan79, 80] for linear programs and is not a combinatorial one. A combinatorial but pseudopolynomial algorithm is proposed by Cunningham [Cunningham85a], where the submodular function / is integer-valued and the required running time is polynomial in \E\ and m a x | / ( X ) | but not in log(max|/(X)|). [Combinatorial strongly polynomial algorithms for minimizing submodular functions were devised by Iwata, Fleischer and Fujishige [IFF01] and Schrijver [SchrijverOO] in 1999 independently and in different ways based on Cunningham's framework [Cunningham84], [Cunningham85a]. See Section 14 for more details.] For special classes of submodular functions the problem of minimizing a submodular function can be reduced to network optimization problems

218

IV. SUBMODULAR ANALYSIS

such as the minimum cut problem [Picard76] and the maximum-weight stable-set problem in a bipartite graph [Billionnet + Minoux85]. We show a "practical" algorithm for minimizing submodular functions based on the following lemma (see (3.22)). Lemma 7.3: min{f(X)

\XeV}

= max{y(£) | y € P ( / ) , y < 0 } .

(7.2)

Consider a special quadratic programming problem over the base polyhedron B(/) described as Minimize ||x||2 (= ^

x(e)2)

(7.3a)

e£E

subject to x £ B(/)

(7.3b)

(see [Fuji80b]). In the present subsection we assume that the underlying totally ordered additive group R is the set of reals or rationals. (In Chapter V we will consider in detail a class of nonlinear optimization problems over the base polyhedron which includes Problem (7.3).) Let x* be the optimal solution of Problem (7.3) and define y*

= x* A 0 (= (min{x*(e), 0} | e € E)),

(7.4)

A- = {e | e € E, x*(e) < 0},

(7.5)

Ao = {e\eEE,

(7.6)

x*{e) < 0}.

Lemma 7.4: y* in (7.4) is a maximize! of the right-hand side of(7.2). Also, A- is the unique minimal minimizer of f and AQ is the unique maximal minimizer of f. (Proof) Because of the optimality of x* we have VeeA_:

dep(x*,e)CA_,

(7.7)

VeeAo:

dep(x*,e) C Ao.

(7.8)

From (7.4)~(7.8), we have A_, Ao £ V and f(A_) = f(Ao) = y*(E) (= x*(A_) = x*(A0)).

(7.9)

7.1. Unconstrained Submodular Programs

219

Since for any X E V and any y E P ( / ) with y < 0 we have f(X) > y{E), it follows from (7.9) that y* is a maximizer of the right-hand side of (7.2) and A- and AQ are minimizers of / . Moreover, for any X E T> such that Ici-, (7.10) f(X) > x*(X) > x*(A.) = f(A.) and for any X E V such that Ao C X, f(X) > x*(X) > x*(A0) = /(Ao).

(7.11)

From (7.9)~(7.11), A_ is the unique minimal minimizer of / and AQ is the unique maximal minimizer of / . Q.E.D. Let c\ < C2 <

< cp be the distinct values of x*(e) (e € E) and define

Bi

=

{e | e € E, x*(e) < ct}

50

=

0.

(i = 1,2,

,p),

(7.12) (7.13)

By a similar argument as (7.7)~(7.9) we see that 0 = Bo C S i C

C BP = E

(7.14)

is a chain of T> and that f(Bl)

= x*(Bl)

(i = 0 , l , - - - , p ) .

(7.15)

Consequently, we have C =

f

i

^ ~f ^

l

)

(i = l , 2 , . . . l P ) .

(7.16)

Problem (7.3) is to find the minimum-norm point in B ( / ) . For simplicity, let us suppose B(/) is bounded, i.e., V> = 2E. A solution algorithm for the minimum norm-point problem is proposed by P. Wolfe [Wolfe76] (also see [von Hohenbalken75]). We can adopt Wolfe's algorithm for Problem (7.3). We express a simplex S in HE by the set of its extreme points.

An algorithm for finding the minimum-norm point in B(/) Step 1: Let x* be any extreme point of B(/). Put S <— {x*}.

220

IV. SUBMODULAR

ANALYSIS

Step 2: Using the greedy algorithm, find a minimum-weight base x of B(/) with respect to the weight function x*. If (x*, x — x*) = 0 , then stop (x* is the minimum-norm point). Otherwise put S <— S U {x}. Step 3: Find the minimum-norm point Xo in the affine space generated by S. If xo is in the relative interior of the simplex S, then put x* <— xo and go to Step 2. Otherwise let S' (Z S be the unique minimal face of simplex S which has the nonempty intersection with the line segment [x*,xo]- Put x* <— the intersection point of the face S' and the line segment [x*, XQ] , and also put S <— S'. Go to the beginning of Step 3. (End) Step 3 is consecutively repeated at most \E\ — 1 times, since the cardinality of S monotonically decreases. Each simplex S available when executing Step 2 uniquely determines x* lying in the relative interior of simplex S and the norm ||x*|| monotonically decreases every time x* is renewed. Therefore, all the simplices S which we encounter during the execution of the algorithm are different, so that the algorithm terminates after a finite number of steps. Example: As an illustrative example, consider the minimum-cut problem for the two-terminal network M = (G = (V, A), s + , s~, c) shown in Fig. 7.1, where V = {s+, s~, 1, 2, 3}. The numbers attached to the arcs denote the capacities c(a) (a € A). The problem is to find a cut U C V with s + € U and s~ ^ U which minimizes its capacity YsaeA+u c(a)> where A+U is the set of arcs leaving U. Putting V* = V — {s + , s~}, we define a submodular function / : 2V* —> R as follows. For each W C V*,

f(w)=

Y,
(7-17)

a€A+{s+}

The second constant term is for the normalization, /(0) = 0. Now, the minimum-cut problem is reduced to the problem of minimizing the submodular function / : 2V* —> R. Let us start with the extreme point x* of B(/) corresponding to the sequence (1, 2, 3) of the vertices in V*, i.e., x*

(= (x*(l), x*(2), x*(3))) = (0,10, -20).

(7.18)

7.1. Unconstrained Submodular Programs

221

Figure 7.1: An example of a flow network. Next, in Step 2 we find by the greedy algorithm the minimum-weight base x of B(/) with respect to weight x*, which is the extreme point of B(/) corresponding to the vertex sequence (3,1, 2), i.e., x = (-30,0,20).

(7.19)

The minimum-norm point XQ in the line through the two points of (7.18) and (7.19) is given by

X =

°

=

2^ 0 ' 10 '- 20 ) + 4 ( ~ 3 0 '°' 2 0 ) ^-(-270,170,-160).

(7.20)

Since 0 < ^|, |r, we put x* <— xo and find a new extreme point, denoted by x, of B(/) corresponding to the vertex sequence (1, 3,2) determined by x* (= XQ in (7.20)). x is given by x = (0,0,-10).

(7.21)

The minimum-norm point XQ in the two-dimensional afline space generated by the three points of (7.18), (7.19) and (7.21) is given by x0 = - | ( 0 , 1 0 , -20) + ^(-30,0,20) + ^ ( 0 , 0 , -10).

(7.22)

222

IV. SUBMODULAR

ANALYSIS

Since — ^ < 0 < ^, ^ , the minimum-norm intersection point of the present simplex formed by points of (7.18), (7.19) and (7.21) and the line segment between points of (7.20) and (7.22) lies in the face formed by points of (7.19) and (7.21). Hence, we discard point (7.18) from the present simplex S and find the minimum-norm point, denoted by XQ again, in the line through points of (7.19) and (7.21). xo = =

* (-30, 0,20) + ^(0,0,-10) (-5,0,-5).

(7.23)

Since the present xo is in the relative interior of the simplex formed by points of (7.19) and (7.21), we put x* <— XQ and go back to Step 2. Now, we find a minimum-weight base x with respect to the weight given by (7.23). Such a base x is given by (7.19) (or (7.21)) and the algorithm terminates. From the minimum-norm base (7.23) we see that UQ = {s+, 1, 2,3} is the unique maximal minimum-cut and U- = {s+, 1,3} is the unique minimal minimum-cut of the network, due to Lemma 7.4. When the algorithm for finding the minimum-norm point in B(/) terminates, we are given the minimum-norm point x* and a set of extreme points Xi (i 6 /) of B(/) such that x* is expressed as a convex combination x* = Y,*iXi

(7-24)

of Xi (i G / ) . For each i G / let V% = (E, ^j) be the poset which corresponds to the distributive lattice V{Xi)

Xi(X)

= {X\XeV,

= f(X)}

(7.25)

with V{xi) = 2Vi, and define a distributive lattice T> by V=(~)2V<.

(7.26)

i€l

Then, T> = V(x*)(={X

\XeV,

x*(X) = f(X)}).

(7.27)

An algorithm for finding the poset V% = (E,
7.1. Unconstrained Submodular Programs

223

E which corresponds to the distributive lattice T> is easily obtained by superimposing the posets V% (i € / ) . For any maximal chain 0 = Ao C A1 C

C A\ = E

(7.28)

of V, the minimum-norm point x* satisfies x*(e) =

f{Aj)

/ ( i j - l )

(7.29)

l ^ - ^ - i I for each e € Aj — Aj-i (j = 1, 2, , q) (cf. (7.16)). Furthermore, all the minimizers of / are given by the interval [A-, AQ]^ of V, where A- and AQ are defined by (7.5) and (7.6). (b) Minimizing modular functions Let /i: T> —> R be a modular function on a simple distributive lattice V> = 2^ with V = (E, ^) and /i(0) = 0. Lemma 7.5: There exists a unique vector v £ R X €P

such that for each

»(X) = J2 Ke)-

(7-30)

eex (Proof) Choose any maximal chain C: 0 = .So C Si C and define a vector v 6 R by »{&) = fi{Sl) - /i(5i_i)

(i = 1,2,

C Sn = E of T> ,

(7.31)

where {ej} = Si — Si-i (i = 1, 2, , n). For any X E T> there exist integers 1 < ji < < j p < n with \X\ = p such that X = {ejk | {ejk} = SJk - 5jfc_i, k = 1, 2, Since /u is a modular function and Sj1 C <Sj2 C

,p}.

(7.32)

C 5j p , we have

fc=i p

= (JL(X U 5jp_i) + E /i((x n 5jfc_i) u S j ^ - i ) + /i(x n 5^-0 fc=2

= !>(%), fc=i

(7-33)

224

IV. SUBMODULAR

ANALYSIS

where use is made of the fact that X U <Sj-p_i = Sjp, X n Sjk-i = I f l Sjk_1 (k = 2, ,p) and X n S^-i = 0. (7.30) follows from (7.31) and (7.33). The uniqueness of v is clear from (7.30). Q.E.D. We see from this lemma that the problem of minimizing the modular function ji:X> —> R, is equivalent to the problem of finding a minimumweight (lower) ideal of the poset V = (E, ^ ) with respect to the weight function v: E —> R. The latter problem was solved by J. C. Picard [Picard76] by reducing it to a minimum-cut problem. Conversely, the reducibility of the minimum-cut problem to a problem of minimizing a modular function was shown by W. H. Cunningham [Cunningham85b]. We first show Picard's approach. Let G{V) = (E, B*{V)) be the graph representing the Hasse diagram of V = (E, ^ ) , i.e., (ei, e^) € B*(V) if and only if e\ covers e^ in V. (The following procedure works if we take, as G(V), any graph G' whose transitive closure coincides with that of G(V), and a (transitively) closed set of G' is an ideal of V.) Consider new vertices s+ and s~ and sets of new arcs S+ = {(s + , e) | e € E, v(e) < 0},

(7.34)

S- = {(e, s - ) | e € E, vie) > 0}.

(7.35)

+

Also define the capacities c(a) of arcs a in B*{V) U S U S~ by f +oo c(a) = { -v(e) [ u(e)

(a£B*(V)) ((s+,e)GS+) ((e,s-)ES-).

(7.36)

Denote this network by J\f = (G = (E,A),s+,s',c), where E = E U + {s+, s~} and A = B*(V) U S U S~ (see Fig. 7.2). For any cut U C E of the network jV (i.e., s+ £ U and s~ <£ U), the capacity of the cut U is given by nc(U) = Y,

{-^(e) \eeE-U,

v[e) < 0}

+ Z]Me) I e € £ n [ / , i/(e) > 0} + J]{ c ( a ) I a

e

B*(V), d+a EU, d'aEE-

U}. (7.37)

Note that for any cut U of finite capacity KC(U), the third term in (7.37) vanishes and U — {s+} is an ideal of V = (E, ^ ) and, conversely, that for any ideal ICEofVIU {s+} is a cut of finite capacity in jV.

7.1. Unconstrained Submodular Programs

225

Figure 7.2: (a) A poset V with weights, (b) Network jV. Furthermore, for any ideal / of V we have KC(/U{S+})

=

£ { - ^ ( e ) | e € £ - J, Ke) < 0} + £ > ( e ) | e e / , z/(e)>0}

=

i/(J) + J ] { - z / ( e ) \e<EE, z/(e) < 0}.

(7.38)

Therefore, minimizing KC(I U { S + } ) for cuts / U {s+} of J\f is equivalent to minimizing u(I) for ideals / of V. See [Picard + Queyranne82] for practical applications of the problem of finding a minimum- or maximum-weight ideal of a poset or a minimumor maximum-weight closed set of a graph. Next, we consider the problem of minimizing a modular function ji:X> —> R from the point of view of submodular analysis. Given a submodular function / : V —> R and a vector x € HE, consider the following problem. It includes the problem of minimizing the submodular function / . (*) Find A € V such that x € df(A) and then find an expression x = xi + x2

(7.39)

226

IV. SUBMODULAR

ANALYSIS

such that x\ is a convex combination of extreme points of df(A) and X2 is a nonnegative linear combination of extreme vectors of Cf(A), the characteristic cone of df (A). Note that the problem of minimizing / is equivalent to that of finding a set A G V such that 0 G df{A). We consider the above problem (*) in the special case when / is a modular function ji: T> —> R with /i(0) = 0. For modular function /x, the subdifferential dfi(A) for each A G V has a unique extreme point v due to Theorem 6.11 and Lemma 7.5, where v is the vector appearing in Lemma 7.5. Hence Problem (*) is reduced to the problem of finding A G T> such that x — v G Cf(A) and of finding a nonnegative linear combination, of ER(^4) given by (6.57), which expresses x — v. The proof of Theorem 6.12 suggests an algorithm as follows. The graph G{V) = {E, B*(V)) represents the Hasse diagram of the poset V = (E, ^ ) . An algorithm for Problem (*) Step 1: Find a nonnegative flow ip: B*(V) —> R+ in G(V) and nonnegative coefficients a(p+) (p+ G E+) and /3{p~) (p~ G E~) such that

E

"(P+)£P++ E

Pip-)rip-+d

(7.40)

(Here, £p+ (p+ G E+) and r]p- (p~ G E~) are vectors in HE denned by (6.54) and (6.55).) For each p+ and p~ such that p+ = p~ G E+ n E1" we choose the values of a(p + ) and (3{p~) so that a(p+)/3(ji~) = 0. Step 2: Construct a network A/" = (G('P),c) with an underlying graph G(V) = (E, B(V)) and a capacity function c: 5 ( P ) -^ R + U {+00} denned as follows. The arc set B(V) of G{V) is given by B(V) = B*(V) U {(e, e') | (e', e) G B*(V)}

(7.41)

and the capacity function c by c(a)-/^ ( a ) C[a) -\+oo

( ae5 *^)) (a=(e,e% (e',e)EB*(V)).

(7 42) '

[

Then find a maximum flow i\>: B(V) —> R+ in J\f from the entrance vertex set E+ — E~ to the exit vertex set E~ — E+ such that O<-0(a)
(aeB(P)),

(7.43)

7.1. Unconstrained Submodular Programs dip{e)={)

227

(e<EE-(E+UE-)),

dip(p+) < a(p+) -di>(p~) < p(p~)

(7.44)

(p+ € E+ - E~),

(7.45)

(p- e E- - E+).

(7.46)

(Here, the boundary operator d is defined with respect to G{V).) Step 3: Put ^(( e ,e')) a(p+)

- ¥J ((e )e '))-^((e,e')) + ^((e', e )) ((e, e') € S*(P)), (7.47) <- a(p+) - di>(p+) (p+ € E+ - E~), (7.48)

/3(p") ^

^(p-) + ^(p") ( p - e ^ - - ^ ) .

(7.49)

Then find A e T> such that (i) cp(a) = 0 (oeA-(A)),

(7.50)

(ii) a(p+) = 0 (p+ € £;+ n A),

(7.51)

(m) ^(p-) = o

(7.52)

(P-G.E;--^).

(End) For any A 6 V satisfying (7.50)~(7.52) we have x 6 d[i(A) and x is expressed as

x = I / + J2 a(P + )^ P ++ E p+£B+

p-€E-

/3(P~)»7p-+

Z

¥>(«)Ca,

(7-53)

aeB*(V)

using the obtained a, /? and ip. Step 1 can be carried out by the breadth-first search and requires linear time. Step 2 is performed by any maximum flow algorithm. Any minimum cut U C E obtained by the maximum flow algorithm in Step 2 gives a desired A G V to be found in Step 3 as A = E - U. From (7.50)^(7.52), x — v in (7.53) is a nonnegative linear combination of ER(yl). When x = 0, the above algorithm solves the problem of minimizing the modular function \i. In this case A € T> found in Step 3 is a minimizer of fj, since 0 G d/i,(A). Let us consider the problem of minimizing a modular function /x: T> —> R, from a polyhedral point of view. Denote by P(X>) the convex hull of all the characteristic vectors xx (X £T>).

228

IV. SUBMODULAR ANALYSIS

Lemma 7.6: For any A £ V, the inequalities of all the facets of the polyhedron P(T>) which include the vertex XA are given by x(e) > 0 (e£E+ - A), (7.54) x(e) - x(e') < 0 (e covers e' in V and either e, e' G A or e, e' (£ A), (7.55) x(e) < 1 ( e e r n A ) . (7.56) (Proof) The lemma follows from the fact that the dual cone (of the tangent cone) of P(V) at \A (regarded as the origin) is the characteristic cone C/I(A) of dfi(A). Notice the one-to-one correspondence between the set of facets (7.54)~(7.56) and the set ER(^4) of the extreme rays of C^A) given by (6.57) with (6.54)^(6.56). Q.E.D. Corollary 7.7: All the facet inequalities of P(D) are given by x(e) > 0 (e € E+),

(7.57)

x(e) - x(e') < 0 (e covers e in V),

(7.58)

x(e) < 1 (eg E~).

(7.59)

The problem of minimizing the modular function ji: V —> R is reduced to the problem of minimizing the linear function

(^x) = 5>( e )x(e)

(7.60)

e€E

over P(£>), where v is the vector appearing in Lemma 7.5. Therefore, it follows from Lemma 7.6 that A E T> is a minimizer of JJ, (or \A is a minimizer of (v, x) over ~P(T>)) if and only if the vector —v is expressed as a nonnegative linear combination of vectors £p+ (p+ € E+ — A), (a (a € B*(V) - A"(A)) and r\v- (p~ € E~ C\A) which are coefficient vectors of (7.54)~(7.56), where inequalities (7.54) should be considered in the form of-x(e) < 0 (eeE+ -A).

7.2. Submodular Programs — Constrained Optimization

7.2. Constrained Submodular Programs

229

We consider the problem of minimizing a submodular function / : D —> R with constraints on the domain D of / and discuss some other related problems. [Here we assume that R is the set of rationals or reals.]

(a) Lagrangian functions and optimality conditions Suppose that a sublattice Do of D is given. We say that a vector a E R is normal to Do at A E Do if for each X E Do a(X) - a(A) < 0.

(7.61)

The following theorem characterizes the minimizers of / when the domain of / is restricted to a sublattice of D . T h e o r e m 7.8 (cf. [Rockafellar70, Theorem 27.4]): Let f:V - R be a submodular function and Do be a sublattice ofV. Then, for A E VQ we have (7.62) f(A) = m i n { / p O | X E Do} if and only if there exists a subgradient a € Of (A) such that —a is normal to D o at A. (Proof) The "if" part: From the assumption, we have for any X E VQ 0
(7.63)

from which (7.62) follows. The "only if" part: Define a modular function IIQ:VQ —> R by m>(X) = 0

(XG D o ).

(7.64)

Then, by the assumption A is a minimizer of /o = f + IIQ:VQ —> R. It follows from Theorem 6.8 and Lemma 7.1 that 0 G dfo(A) = df{A) + dfjio(A).

(7.65)

Hence, there exists a vector a € Of (A) such that -a G dfiO(A). From (7.64), (7.66) implies that —a is normal to Do at A.

(7.66) Q.E.D.

230

IV. SUBMODULAR ANALYSIS

Next, consider a constrained minimization problem for which the "feasible region" T>Q in (7.62) is denned by a set of equations. Suppose that we are given submodular functions ff-V —> R (i = 0,1, ,m) and that the minimum value of fi for each i = 1, 2, , m is known and equal to U{. The feasible region VQ is given by V0 = {X | X € V, Vi e {1, 2,

, m}: ft{X) = at}.

(7.67)

Here, we assume VQ / 0. From Lemma 2.1, VQ is a sublattice of V. Let us consider the following constrained minimization problem: Minimize fo(X)

(7.68a)

subject to fi(X) = oti (i = 1, 2,

, m).

(7.68b)

Define a function L : R ™ x P ^ R b y m

L(X, X) = fo(X) + ] T Xi(fi(X) -

ai)

(7.69)

for A = (Ai, A2, , Am) € R+ and X £ V. We call L the Lagrangian function associated with Problem (7.68). Also, we call A € R/J2 an optimal Lagrange multiplier if min{L(A,X) \ X <E V}

(7.70)

is equal to the optimal value of the objective function of Problem (7.68). Theorem 7.9 (cf. [Rockafellar70, Theorem 28.3]): For Problem (7.68), (1) A = (Ai,

, Am) is an optimal Lagrange multiplier and

(2) X is an optimal solution of Problem (7.68) if and only if (i)

k(A i r -,A T O )eR™

(ii)

X is a feasible solution of Problem

(7.68) and

0 G dfo(X)

Xmdfm(X).

(iii)

+ XidhiX)

+

+

7.2. Constrained Submodular Programs

231

(Proof) First, note that for each i = 1, , m, dfi(X) and dfo{X) has the same characteristic cone, since the characteristic cone is determined by the underlying distributive lattice T> alone. Therefore, dfo{X)

= dfo(X)

+ O+dfl(X)

(i = l,2,---,m).

(7.71)

Hence, from Theorem 6.8 and Lemma 7.1, (iii) is equivalent to OGdxL(X,X),

(7.72)

where dxL(X, X) denotes the subdifferential of the submodular function L(X,-): P ^ R a t l The "if" part: From (i)~(iii) and (7.72) we have min{L(A, X) \ X E V} = L(X, X) = fo(X),

(7.73)

while we have for any feasible solution Y min{L(A, X) \ X € V} < L(X, Y) = fo(Y).

(7.74)

Hence (1) and (2) follow. The "only if" part: From (1) and (2) we have X e Vo,

(7.75)

(7.76) min{L(A, X) \ X E V} = fo(X) = L(A, X). Therefore, we have (7.72), or (iii). (i) and (ii) trivially follow from (1) and (2). Q.E.D. The above proof almost parallels that of the corresponding theorem for ordinary convex functions ([Rockafellar70, Theorem 28.3]). It should, however, be noted that the above proof heavily depends on the results in Section 6.2, especially Theorem 6.8. Define a function p: R ^ R U {+00} by p{u) = min{/opO \XeV,Vie{l,---,m}--

fi{X)-at
(7.77)

where u = (u\, , um) € R+. We define p(u) = +00 for u for which there exists n o I e P such that fi{X) — oti < u-i for alii = 1, , m. We call p the perturbation function associated with Problem (7.68).

232

IV. SUBMODULAR

ANALYSIS

Theorem 7.10: For the perturbation function p associated with Problem (7.68) we have for each A e R ^ min{p(u) + Aitti +

\- Xmum

\ u G R" 1 }

= min{L(A,X) | X G £>}.

(7.78)

(Proof) Suppose that X G V is a minimizer of L(X, X) in X G V. Then, from the definition of p(u), L(X, X) = p(u) + Aiui + where ui = fi{X) — a\ (i = 1,

+ Xmum,

(7.79)

, m). Hence we have

min{p(u) + Aitti + \- \mum < min{L(A,X) \ X eV}.

\ u G R™} (7.80)

On the other hand, suppose that u 6 R™ is a minimizer of p(u) + Ai^i + ' + ^n>um in u G R™. Then, there exists an XQ G P such that p(u) = / o (Xo), fi(X0)-at
(7.81)

(i = l , - - . , m ) .

(7.82)

Since A G R ^ , we have from (7.81) and (7.82) p{u) + Aiui +

+ Xmum > L(X, Xo).

(7.83)

Therefore, h AmMm | u G R" 1 } min{p(u) + Aitti H >min{L(A,X) | X € P } . The present theorem follows from (7.80) and (7.84).

(7.84) Q.E.D.

In the proof of Theorem 7.10 we have already shown the following. Theorem 7.11: For each A G R + , (a) if X G V is a minimizer of L(X, X) in X G V, then u = (ui, given by ul = fl{X)-ai (i = l,...,m) is a minimizer ofp(u) +

+

+ Xmum

in u G R™;

, um) (7.85)

7.2. Constrained Submodular Programs

233

(b) if u = (ui, um) is a minimizer of p[u) + \\U\ + u G R " \ then there exists an X G V such that p(u) = fo(X), fi(X)-ai
(i = l,---,m),

+ Xmum

in

(7.86) (7.87)

and X is a minimizer of L(A, X) in X G T>. Here, for each i = 1, ,m, (7.87) holds with equality if \ > 0, while A, = 0 if (7.87) for i holds with strict inequality. Furthermore, we have Theorem 7.12 (cf. [Rockafellar70, Theorem 29.1]): A vector A e R+ is an optimal Lagrange multiplier of Problem (7.68) if and only if min{p(u) + Aitti -\

h \mum

| u G R + } = p(0).

(7.88)

(Proof) (7.88) means that the zero vector 0 G R"J is a minimizer of p(u) + Aiiii + + Xmum in u G R™. Hence, the present theorem follows from Theorems 7.9~7.11. Q.E.D. We see from Theorems 7.11 and 7.12 that if A G R!j? is an optimal Lagrange multiplier, then any A G R™ such that A < A is also an optimal one. Note that (7.87) for i such that Ui = 0, holds with equality since fi{X) - a% > 0 for any X G V. An algorithm for finding an optimal solution and an optimal Lagrange multiplier for Problem (7.68) is given as follows. For simplicity, we consider the case when m = 1. An algorithm for solving Problem (7.68) (with m = 1) Step 1: Let a,o be an upper bound of /o such that ao > fo{X) for all X G P (with strict inequality) and let ao be a lower bound of / o Also let a\ be an upper bound of / i such that > a>\. Put A <— (ao — «o)/(ai — ct\). Step 2: Put X <- a minimizer of L(X,X) X (EV.

= fo(X) + \(fi(X)

- ax) in

Step 3: If fi(X) — = 0, then stop (X is an optimal solution and A is an optimal Lagrange multiplier). Otherwise, put A <— (ao — fo(X))/(f\(X) — ct\) and go back to Step 2.

234

IV. SUBMODULAR ANALYSIS

(End) The validity of the above algorithm follows from Theorems 7.8, 7.11 and 7.12. The case when m > 1 can also be treated by the algorithm by setting / i ^ / i + h +

" + fm- If mm{fi(X)

+ f2(X) +

+f m ( X ) | X €

V} ^ a\ + ct2 + + am, then there is no feasible solution. The upper bounds of the submodular functions required in Step 1 are obtained by adapting (3.89). When fi (i = 0,1) are integer-valued, the above algorithm terminates after repeating the cycle of Steps 2 and 3 at most 2min{ao — ach^i — " l } times, where upper bounds a, (i = 0,1) and a lower bound ao are chosen to be integral. For each A € R™ denote by C(X) the set of minimizers of the Lagrangian function L(A, X) = fo(X) + Ai/ipO + + Xmfm(X) (7.89) in X € V. £(\) is a sublattice of V. Because of the fmiteness character of the problem there are a finite number of distinct £(A)'s (A € R+). The structure of these £(A)'s is closely related to the concept of principal partition ([Kishi + Kajitani68], [Iri71], [Tomi76], [Narayanan74], [Puji78c], [Iri79], [Nakamura + Iri81], [Iri84], [Nakamura88a], [Tomi + Fuji82]), which will be discussed in the subsequent subsection.

(b) Related problems

We discuss other problems related to submodular programs. (b.l) The principal partition [Iri79], [Nakamura+Iri81], [Tomi+Fuji82] Let us consider the Lagrangian function L(X, X) of (7.69) for the case when m = 1 and we suppress the term a\ in L(X,X). We suppose that (Z>j,/j) (i = 0,1) are submodular systems on E, that T>\ is a Boolean lattice (i.e., T>\ = T>\ (= {E — X | I g P i } ) ) , and that /i is monotone nonincreasing. The monotonicity of /i plays an essential role in the following argument. Define V = V0DV1.

Consider the Lagrangian function L(X,X) = fo(X) + Xf1(X)

(7.90)

7.2. Constrained Submodular Programs

235

for A > 0 and I £ P . We also define L(A, X) for A < 0 as L(X,X) = fo(X) + Xf1#(X),

(7.91)

where f\* is the dual supermodular function of / i , i.e., f\*(X) = fi(E) — fi{E - X) (X E P i ) . Note that for any A e R L(X,X) is a submodular function in X G T>. For each A G R let £(A) be the set of minimizers of L(X,X) in X € V. £(A) is a sublattice of V. Theorem 7.13 ([Tomi + Fuji82]): Define

C = |J C(X).

(7.92)

AeR JC* is a sublattice ofD. More precisely, for any X and A' with A < A' and for any X € £(A) and X' € C(X'), we have xnx'eC(X),

iui'e£(A').

(7.93)

(Proof) If A = A', (7.93) holds. Case 1: 0 < A < A' For any X G £(A) and X' G £(A') we have L(X,X) + L(X',X') = fo(X) + A/i(X) + fo{X') + A'/ipO

> fo(x u x') + A'/I(X u x') + / fl (in x') + A/I(X n x') +(\'-\)(f1(xnx')-fl(x)) > L(\', X U X') + L(\, X D X')

(7.94)

due to the submodularity of /o and / i and the monotonicity of / i . It follows from (7.94) that L(X',XUX') = L(X',XI), (7.95) L(X,XDX') = L(X,X) (and /i(X) = / i ( X f l l ' ) ) . We thus have (7.93). Case 2: A < 0 < A'

(7.96)

236

IV. SUBMODULAR

ANALYSIS

Similarly as (7.94), we have L(X,X) + L(X',X')

= fQ(X) + \h*{X) + fo(X') + X'h(X')

> fo{x u x1) + x'h(x u x') + fo(x n x1) + Xfx*{x n x') +x'(h(x n x') - h{x)) + x{h#{x) - h*{x n x'))

>L(A',IUl') + i(A,lnl').

(7.97)

From this we have (7.93). Case 3: X < X' < 0

Similarly, we have L(X,X) + L(X',X')

= fo(X) + Xh#{X) + fo(X') + X'f^iX1)

> fo(x ux1) + x'f\#{x u x1) + fo(x nx') + xf\#{x n x')

+(A'-A)(/1#(I')-/1#(IUI')) >L(X',XUX') + L(X,XnX').

(7.98)

Hence we have (7.93).

Q.E.D.

Denote by S+(X) the unique maximal element of £(X) and by S~(X) the unique minimal element of C(X) for each A € R. Theorem 7.14: For any X and X1 with X < X1 we have S+(X)CS+(X'),

S-(X)CS-(X').

(7.99)

(Proof) For any X G £(A) and X' £ C(X') we have from Theorem 7.13 XUS+(X')<EC(X'),

s~(x)nx'

<EC(X).

(7.100)

This implies X C S+ (A') (X e £(A)) and 5~(A) C X' (X1 e C(X')). Hence we have (7.99). Q.E.D. We further suppose that f\ is monotone decreasing. Then, for a sufficiently large A > 0 we have C{X) = {E} and for a sufficiently small A < 0, C(X) = {0}.

7.2. Constrained Submodular Programs

237

Theorem 7.15: Suppose that f\ is monotone decreasing. exists a sequence of reals Ai
Then, there

(7.101)

with p < \E\ such that the distinct C(X) (A € R) are given by C(\i) {S+(Xt)}

(i = l,2,---,p),

(= { 5 - ( A i + i ) »

{5-(A!)} (= {0}), For any i € {1,2,

(7.102)

(i = l , 2 , . . . , p - l ) , {S+(\p)}

(= {E}).

(7.103) (7.104)

,p — 1} and any A such that A, < A < Aj+i we have £(A) = {5 + (A,)} = {5"(A i + 1 )}.

(7.105)

Also,

(Proof) Because of the flniteness character there exist finitely many distinct £(A) (A € R). Choose any A € R. If £(A) contains more than one element, there exist some distinct X, X' G £(A) with X c X' and

fo{X) + A/ip0 = /o(X') + A/iOT-

(7-107)

Since fi(X) > fi(X') by the monotonicity assumption, the value of A is uniquely determined from (7.107). If £(A) contains only one element, then from the finiteness character there exists an open interval (A', A") such that A e (A', A") and £(A) = £(A) (Ae(A / ,A / / )). (7.108) It follows that there exists a finite sequence of reals Ai
(7.109)

such that distinct £(A) (A € R) are given by C(Xi) £(A)

(i = l , 2 , - - - , p ) )

(7.110)

(A€(Ai,A i + i), i = 0,l, . . . , p ) ,

(7.111)

238

IV. SUBMODULAR

ANALYSIS

where Ao = —oo, Ap+i = +00, £(A)'s are the same in each interval (Aj, Aj+i) (i = 0,1, -,p), \C(Xi)\ > 2 (i = 1, 2, ,p) and |£(A)| = 1 (A € (A,, Xi+1), i = 0,1, ,p). Moreover, for each i = 1, 2, ,p, because of the finiteness character there exists a (sufficiently small) positive number e such that £(Ai - e) C £(Ai),

(7.112)

£(Ai + e)C£(Ai).

(7.113)

Since /1 is monotone decreasing, we have from (7.112) and (7.113) S-(\i)eC(Xi-e),

(7.114)

5+(A,)€/:(A i + e).

(7.115)

From (7.114) and (7.115) we have (7.103)~(7.106). Also, since the set of the quotients S+(Xi) — 5 + (Aj_i) (i = 1, 2, ,p) with S+(Xo) = 0 is a partition of E into nonempty subsets of E due to Theorem 7.14 and (7.103)~(7.106), we h a v e p < \E\. Q.E.D. The Aj (i = 1, 2, ,p) in (7.101) are called critical values for the pair of submodular systems (T>o, fo) and (T>i,f{). Denote So = (^O)/o) a n d Si = (Pi, /1). Submodular systems Sj (i = 0,1) are decomposed according to the distributive lattice £* = UAGR ^(^) a s fouows- Choose any maximal chain C: 0 = Ao(ZA1G---(ZAk =E (7.116) of C* and then decompose Sj (i = 0,1) into their minors Si-Aj/4/-i

G? = 1,2,

,

(7.117)

where Sj Aj/Aj-i is the set minor of Sj obtained by restricting Sj to Aj and contracting A7-1. Such a set of decompositions of Sj (i = 0,1) is called the principal partition of the pair of Sj (i = 0,1). By the poset on the partition , k} of E which is uniquely determined by C* (see {Aj — Aj-i I j = 1, 2, Section 3.2.a), the corresponding poset structure is denned on the set of minors (7.117) for each i = 0,1. We can show that the decompositions (7.117) do not depend on the choice of a maximal chain in C* ([Nakamura + Iri81], [Tomi + Fuji82]), due to Theorem 7.17 shown below. L e m m a 7.16: Let \i; V$ —> R be a modular function on a distributive lattice Vo C 2E with Q,E G Vo. We have fi(X) = 0 for all X £ V if

7.2. Constrained Submodular Programs and only if fi(Si) ® = S0(lS1c---(zSk

= 0 (i = 0 , 1 , = E ofV0.

239 ) for an arbitrary

maximal

chain

(Proof) This lemma immediately follows from Lemma 7.5 and Corollary 3.10. Q.E.D. Theorem 7.17 [Nakamura+Iri81] (also see [Tomi+Fuji82]): For a submodular system (V, / ) on E suppose that f is modular on a sublattice VQ of V with 0, E € VQ. For a maximal chain 0 = So C Si C C Sk = E of T>o consider minors (T>, / ) Si/Si-i (i = 1,2, , A;). Then, the set of these minors is independent of the choice of a maximal chain of Do. (Proof) For a vector x £ HE the following two statements are equivalent (see Lemma 3.1): (i) x 3 * - 3 * - 1 i s a b a s e o f (V, / ) Si/S^i

for e a c h i =

l,2,---,k.

(ii) x is a base of (V, f) and f(Si) — x(Si) = 0 for each i = 0,1,

, k.

Since f — x: T>Q —> R. is modular on X>Q, we see from Lemma 7.16 that (ii) is also equivalent to (iii) x is a base of (V, f) and f(X) - x(X) = 0 for each X £ Vo. It follows from the equivalence between (i) and (iii) that the set of minors (T>, / ) Si/Si-i (i = 1, 2, , k) does not depend on the choice of a maximal chain of P o Q.E.D. The concept of principal partition was originated by G. Kishi and Y. Kajitani [Kishi + Kajitani68] for graphs, where So is a graphic matroid and Si = (2E, - 1 ) with uniform modular function -1(X) = -\X\ (X C E). It was generalized to a pair of graphs [Ozawa74]; to a pair of a matrix and a uniform modular function [Iri69b,71]; to a pair of a matroid and a uniform modular function [Narayanan74], [Tomi76] (also [Bruno + Weinberg71]); to a pair of a polymatroid and a positive modular function [Fuji78c], [Fuji80b]; to a pair of polymatroids [Iri79]; to a pair of a submodular system and a modular function [Fuji80c]; and to a pair of submodular systems [Nakamura + Iri81], [Tomi80d], [Tomi + Fuji82]. The concept has effectively been applied to electrical network analysis [Iri + Tomi75], [Narayanan74], network flows [Fuji80b], structure and scene analyses [Sugihara82,86], the determination of the concept structure [Sugihara + Iri80], etc. (Also see [Iri79], [Iri83], [Iri + Fuji81], [Tomi + Fuji82]).

IV. SUBMODULAR ANALYSIS

240

Example b.l: The principal partition of a graph Consider a graph G = (V,E) shown in Fig. 7.3. Let f0: 2E -> Z+ be the rank function r of the graph and let f\: 2E -> Z+ be the modular function which corresponds to the negative of the uniform vector 1 = (1,1, ,1) E RE. WehaveL(A,X) = r(X)-X\X\. The set (7.117) of minors S0-A,-/Aj_i (j = 1, 2, , 10) together with the poset structure is given in Fig. 7.4. The principal partition shown in Fig. 7.4 is the decomposition in the sense of Tomizawa and Narayanan. Note that each minor S0-Aj/Aj-i, considered as a polymatroid with a graphic rank function, has the uniform base Xj) € R A * ^ - 1 with the critical value Xj corresponding u\. = (Xj,Xj, to'the minor. The direct sum of these bases uXj of minors S o Aj/Aj-i , 10) is a base of the original graphic polymatroid and, decom(j = 1, 2, posing the exchangeability graph associated with this base into strongly connected components, we have the principal partition shown in Fig. 7.4 (see [Fuji80c]).

Figure 7.3: A graph. For each graph Gj = (Vj, Ej) representing the minor S o Aj/Aj-i 1, 2, , 10), the critical value Xj is given by j

= (\Vj\-l)/\Ej\,

(j = (7.118)

7.2. Constrained Submodular Programs

241

so that 1/Aj can be regarded as the "density" of Gj = (Vj,Ej). A minor with the smallest critical value gives a subgraph of G with the maximum density (cf. [Goldberg83]).

Figure 7.4: The principal partition of the graph in Fig. 7.3. The principal partition of a graph G = (V, E) in the sense of Kishi and Kajitani is the partition of E into three parts E+, Eo, E- such that G+ = G x E+, Go = (G - E+)/E-, and G_ = G E- have the critical values greater than ^, equal to ^, and less than ^, respectively. Based on this tri-partition, we can determine the topological degree of freedom of the (electrical) network represented by G. The topological degree of freedom is the minimum number of current variables and voltage variables whose values uniquely determine the value of the current variable or the voltage variable of each arc through Kirchhoff 's current and voltage laws. The topological degree of freedom of G is given by min{r(X) + \E — X\ — r*{E - X) I X C E} in terms of the rank function r of G. Note that \E - X\ - r*{E - X) is the nullity of the contraction G/X and that r(X) + \E-X\- r#(E -X)= 2r{X) - \X\ + \E\ - r(E). Therefore, the problem is reduced to minimize 2r(X) — \X\ or r{X) — \\X\ and the topological

242

IV. SUBMODULAR ANALYSIS

degree of freedom of G is equal to the sum of the rank of G_, the nullity of G+, and the rank (or nullity) of Go (see [Iri68,69], [Kishi + Kajitani68] and [Ohtsuki + Ishizaki + Watanabe68]). Kishi and Kajitani's tri-partition also furnishes the solution of Shannon's switching game (see [Edm65b], [Bruno + Weinberg71]). Shannon's switching game is described as follows. Two players play on a graph G with a special reference edge eo. Alternately choosing one edge of G, one player, called a short player, tries to construct a cycle consisting of the reference edge eo and some of the arcs already chosen by the player, and the other player, called a cut player, tries to construct a cutset (cocircuit) containing eo- The short (or cut) player wins if he succeeds in constructing such a cycle (or cutset). If eo belongs to G_ (or G + ), then the short (or cut) player can win; and if eo belongs to Go, the first-move player can win (see [Bruno + Weinberg71]).

Example b.2: The principal partition of a polymatroid induced by a multi-terminal network Consider a capacitated single-source multiple-sink network J\f = (G = (V, A), c, s+,S~) shown in Fig. 7.5, where the label attached to an arc a € A denotes its capacity c(a), s+ is the source and S~ = {s^ \ i = 1, 2, , 6} is the set of the sinks.

Figure 7.5: A capacitated single-source multiple-sink network.

7.2. Constrained Submodular Programs

243

For each feasible flow ip from s + to S~ in J\f define d~ip € R S by d-
= -dtp(sr)

( i = 1,2,

.

(7.119)

Then the set of vectors d~