SIAM J. DISCRETE MATH. c© 2007 Society for Industrial and Applied MathematicsVol. 21, No. 2, pp. 349–360

ON MAXIMUM COST KT,T -FREE T -MATCHINGS OF BIPARTITEGRAPHS∗

MARTON MAKAI†

Abstract. Frank examined the maximum Kt,t-free t-matching problem of simple bipartitegraphs. As the C6-free 2-matching problem is NP-hard (Geelen), this is a promising generalizationof restricted 2-matchings. Given an arbitrary family T of Kt,t-subgraphs of the underlying graph,a T -free t-matching is a subgraph of maximum degree at most t that contains no member of T .We show that the maximum size T -free t-matching problem also admits a nice min-max formula.Given an integer cost function on the edge-set which is vertex-induced on any member of T , we alsoshow an integer min-max formula for the maximum cost of T -free t-matchings. As the maximumcost C4-free 2-matching problem is NP-hard (Kiraly), we cannot expect a nice characterization ingeneral.

Key words. submodularity, restricted matchings, integer polyhedra

AMS subject classifications. 05C07, 05C70, 05C65, 90C10, 90C27, 90C46, 90C47, 90C57

DOI. 10.1137/060652282

1. Introduction. Throughout the paper, we work with the finite simple bipar-tite graph G = (V = A ∪ B,E), and t ≥ 2 will be an integer. For a graph H, V (H)and E(H) denote, respectively, its set of vertices and edges. If H is a subgraph of G,then its color classes will be denoted by A(H) = A ∩ V (H) and B(H) = B ∩ V (H).(The same notation will be used also in some other situations.) If H is a graph(hypergraph), X ⊆ V (H), then H[X] denotes the subgraph (subhypergraph) of Hinduced by X, and |E(H[X])| is denoted by iH(X). For X,Y ⊆ V (H), X ∩ Y = ∅,δH(X,Y ) = δH({X,Y }) denotes the set of edges of H going between X and Y , δH(X)stands for δH(X,V −X), and for the singleton {v}, we use δH(v) rather than δH({v}).If δ is replaced by d, then it denotes the cardinality of the corresponding set. In thesenotations, the graph is sometimes replaced by its edge set, or if the graph (edge set) isclear from the context, its notation is omitted. For a function g : V → Z and X ⊆ V ,we use g(X) =

∑v∈X g(v); we do not distinguish subsets of V and their characteristic

functions, nor do we distinguish vectors and functions on the same ground set.For f, g : V → Z, f ≤ g, an (f, g)-factor of G is a subgraph H of G such that

(s.t.) f(v) ≤ dH(v) ≤ g(v) for every v ∈ V . The (0, g)-factors are called g-matchings,and the (g, g)-factors are called g-factors. In the literature, a matching may consistof multiple copies of an edge; our notion of matching, consisting of subgraphs, isreferred to as simple. Since we deal only with simple graphs and simple matchings,the adjective “simple” is omitted. The problem of searching for a matching with amaximum number of edges is known as the maximum matching problem. Similarly,given a cost function c : E → Z, the maximum cost matching problem is to search for amatching H maximizing its cost c(E(H)). It is known from bipartite matching theory

∗Received by the editors February 17, 2006; accepted for publication (in revised form) November 6,2006; published electronically May 7, 2007. This research was supported by the Hungarian NationalFoundation for Scientific Research, OTKA grants T037547 and TS 049788, by European MCRTNAdonet, contract grant 504438, and by the Egervary Research Group of the Hungarian Academy ofSciences.

http://www.siam.org/journals/sidma/21-2/65228.html†Department of Operations Research, Eotvos University, Pazmany Peter setany 1/C, H-1117

Budapest, Hungary, and Communication Networks Laboratory, Pazmany Peter setany 1/A, H-1117Budapest, Hungary (marci@cs.elte.hu).

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350 MARTON MAKAI

(see, e.g., [10]) that the maximum number of edges of a g-matching in a bipartite graphG is

minZ⊆V

g(Z) + iG(V − Z).

Similarly, a simple formula is known for the maximum cost of g-matchings (g-factors),although it is better to formulate this in polyhedral terms.

Generalizing this, Cunningham and Geelen proposed investigating the maximumC4-free 2-matching problem, i.e., the problem of finding a maximum 2-matching thatdoes not contain a cycle of length four. Hartvigsen obtained a min-max formula by acombinatorial algorithm and introduced the linear program

max x(E),(1.1)

x ∈ RE , 0 ≤ x ≤ 1,(1.2)

x(δ(v)) ≤ 2 for every v ∈ V ,(1.3)

x(E(C)) ≤ 3 for every subgraph C of G isomorphic to C4.(1.4)

Clearly, the integer solutions of (1.2)–(1.4) are exactly the C4-free 2-matchings. Hart-vigsen proved the following integrality result.

Theorem 1 (Hartvigsen [5, 6]). The optimum of the linear program (1.1)–(1.4) isattained on an integer vector, and the optimum of its dual is attained on a half-integervector.

Kiraly sharpened this by stating that the dual has, in fact, integer optimal solu-tions, and he obtained the following theorem by a relatively simple inductive proof.

Theorem 2 (Kiraly [9]). For g : V → {0, 1, 2}, the maximum number of edgesof a C4-free g-matching of the bipartite graph G is

minZ⊆V

g(Z) + iG(V − Z) − c2(G[V − Z]),

where c2(G[V − Z]) denotes the number of C4-components of G[V − Z].The crucial observation in the area was made by Frank. As Geelen proved that the

maximum C6-free 2-matching problem is NP-hard [4], Frank proposed generalizing theC4-free 2-matching problem by forbidding Kt,t subgraphs in t-matchings. His approachis based on the general set-pair covering theorem of Frank and Jordan [3], thus notleading to a combinatorial algorithm. Later, Kiraly also was able to extend his prooffor this case [8].

Theorem 3 (Frank [2] with f ≡ t, Kiraly [8]). For g : V → {0, 1, 2, . . . , t}, themaximum number of edges of a Kt,t-free g-matching of the bipartite graph G is

minZ⊆V

g(Z) + iG(V − Z) − ct(G[V − Z]),

where ct(G[V − Z]) denotes the number of Kt,t-components of G[V − Z].We emphasize that neither of these two approaches is algorithmic. The proof

based on the Frank–Jordan theorem provides a polynomial time algorithm via theellipsoid method [1, 3], but a purely combinatorial algorithm is not known. It may bepossible to extend Hartvigsen’s maximum C4-free 2-matching algorithm for the abovecase, but it would be much more interesting to see an algorithmic approach via theFrank–Jordan theorem.

Frank’s technique yielded another generalization of the problem. A completebipartite graph Kk,l with k + l > t + 1 and k, l ≥ 1 is said to be a large biclique.

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Kt,t-FREE t-MATCHINGS OF BIPARTITE GRAPHS 351

Frank [2] proved a min-max formula for the maximum number of edges of a subgraphH of G which contains no large biclique. Notice that, for t = 1, this is the maximummatching problem and that, for t = 2, this is the K2,2-free 2-matching problem.

The natural question of maximum cost restricted matchings also arises. Kiralynoticed [7] that Geelen’s proof [4] can be modified to show that the maximum costC4-free 2-matching problem is NP-hard. Hence, the more general Kt,t-free t-matchingproblem is also NP-hard. On the other hand, Frank’s approach enables us to handlevertex-induced cost functions, i.e., any cost function c : E → Z for which there existsc′ : V → R s.t. c(uv) = c′(u) + c′(v) for every uv ∈ E [2].

The main purpose of this paper is to approximate better the borderline of tractabil-ity in the maximum cost Kt,t-free t-matching problem by a polyhedral study. We givemin-max formulae for the maximum cost problem for a class of cost functions whichis more general than the class of vertex-induced cost functions.

Suppose that we are given a function g : V → {0, 1, 2, . . . , t} and an arbitraryfamily T of the Kt,t-subgraphs of G, which is called the set of forbidden Kt,t’s. A T -free g-matching (g-factor, (f, g)-factor) is a g-matching (g-factor, (f, g)-factor) thatcontains no member of T . A cost function c : E → Z is said to be T -induced if,for every T ∈ T , there exists cT : V (T ) → R s.t. c(uv) = cT (u) + cT (v) for everyuv ∈ E(T ). In other words, T -induced cost functions are vertex-induced on forbiddenKt,t’s. Formulated in polyhedral terms, our main result is the following.

Theorem 4. Let G be a bipartite graph, f, g : V → {0, 1, 2, . . . , t}, f ≤ g, andlet T be an arbitrary family of Kt,t-subgraphs of G. If c : E → Z is a T -induced costfunction, then the optimum of the linear program

max cx(1.5)

x ∈ RE , 0 ≤ x ≤ 1,(1.6)

f(v) ≤ x(δ(v)) ≤ g(v) for every v ∈ V ,(1.7)

x(E(T )) ≤ t2 − 1 for every T ∈ T(1.8)

and the optimum of its dual are attained on integer vectors.If there is no forbidden Kt,t, then all the integer cost functions are T -induced,

while if T is very dense in G, then the set of T -induced cost functions coincides withthe set of vertex-induced cost functions. It is not hard to see that Theorem 4 impliesthe following for maximum T -free g-matchings.

Theorem 5. Let G be a bipartite graph, g : V → {0, 1, 2, . . . , t}, and let T bean arbitrary family of Kt,t-subgraphs of G. Then the maximum number of edges of aT -free g-matching is

minZ⊆V

g(Z) + iG(V − Z) − cT (G[V − Z]),

where cT (G[V − Z]) denotes the number of T -components of G[V − Z].The proof of Theorem 4 is based on the primal-dual method and on the following

theorem, which characterizes the existence of T -free (l, u)-factors.Theorem 6. For l, u : V → Z, 0 ≤ l ≤ u ≤ t, G has a T -free (l, u)-factor if and

only if, for each X ⊆ A and Y ⊆ B,

l(X) ≤ u(Y ) + iG(X ∪B − Y ) − cT (G[X ∪B − Y ])(1.9)

and

l(Y ) ≤ u(X) + iG(Y ∪A−X) − cT (G[Y ∪A−X])(1.10)

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352 MARTON MAKAI

hold, where cT (G) denotes the number of components of G that are members of T .In the rest of the paper, the proofs of Theorems 4 and 6 are presented. The proof

of the latter result is based on a slightly modified version of the Frank–Jordan theorem.The goal of this generalization is twofold. The first aim is to end up with a proof ofTheorem 6, while the second one is to show that there are possible generalizations ofthe Frank–Jordan theorem where the uncrossing operation is not so apparent.

2. Proof of Theorem 4.Proof of Theorem 4. Let c be a T -induced cost function, and let (y, π, λ, z) ≥ 0,

y : E → Z, π : V → Z, λ : V → Z, z : T → Z, be a (not necessarily optimal) integerdual solution, where πv is associated with the constraint x(δ(v)) ≤ g(v), and λv isassociated with −x(δ(v)) ≤ −f(v). (Note that there always exists a dual solution,say y = c, π = 0, λ = 0, z = 0.)

An edge uv ∈ E is said to be tight if the dual inequality

yuv + πu + πv − λu − λv +∑

T∈T :uv∈E(T )

zT ≥ c(uv)(2.1)

holds with equality. For F ⊆ E and S ⊆ T , we introduce the notations Ftight ={e ∈ F : e is tight}, F0 = {e ∈ F : ye = 0}, F+ = F − F0, S0 = {T ∈ S : zT = 0},and S+ = S − S0. Therefore, the set of tight edges e with ye = 0 is denoted byEtight,0. Moreover, let us choose (y, π, λ, z) so that the vector (w1, w2, w3, w4) definedby w1 =

∑e∈E ye+

∑v∈V (πvg(v)−λvf(v))+

∑T∈T (t2−1)zT , w2 =

∑T∈T (t2−1)zT ,

w3 =∑

e∈E ye, w4 =∑

v∈V (πv + λv) is lexicographically as small as possible.In what follows, we either construct a primal solution which satisfies the com-

plementary slackness conditions with respect to (y, π, λ, z), or we construct a dualsolution (y′, π′, λ′, z′) s.t.

∑e∈E

y′e +∑v∈V

(π′vg(v) − λ′

vf(v)) +∑T∈T

(t2 − 1)z′T

<∑e∈E

ye +∑v∈V

(πvg(v) − λvf(v)) +∑T∈T

(t2 − 1)zT .

First, we need some technical observations.Lemma 7.

(i) E+ ⊆ Etight.(ii) The members of T+ are disjoint.(iii) For any v ∈ V , either g(v) = 0 and δ(v)+ = ∅, or g(v) > 0 and |δ(v)+| <

g(v).(iv) If λv > 0, then |δ(v)+| < f(v).(v) If T ∈ T+ and e ∈ E(T ), then e ∈ Etight,0.(vi) If v ∈

⋃{V (T ) : T ∈ T+}, then g(v) = t.

(vii) If v ∈⋃{V (T ) : T ∈ T+} and λv > 0, then f(v) = t.

(viii) If T ∈ T+, uv ∈ E, u ∈ V (T ), and v /∈ V (T ), then yuv = 0.(ix) At least one of πv = 0 or λv = 0 holds for every v ∈ V .Proof. If any of the above statements does not hold, then we show that (y, π, λ, z)

can be replaced by (y′, π′, λ′, z′) so that the corresponding (w′1, w

′2, w

′3, w

′4) is strictly

smaller than (w1, w2, w3, w4). We define (y′, π′, λ′, z′) only on the coordinates whereit changes compared to (y, π, λ, z).

(i) If e ∈ E+ is not tight, then let y′e = ye − 1, and therefore w′1 < w1.

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Kt,t-FREE t-MATCHINGS OF BIPARTITE GRAPHS 353

(ii) Let T, S ∈ T+ s.t. V (T ) ∩ V (S) �= ∅. Then let z′T = zT − 1 and z′S = zS − 1,π′v = πv + 1 for every v ∈ V (T ) ∩ V (S), and y′e = ye + 1 for every e ∈ E[V (T ) −

V (S)] ∪E[V (S)− V (T )]. Setting a = |A(T ) ∩A(S)| and b = |B(T ) ∩B(S)|, we haveg(A(T ) ∩A(S)) ≤ t(a+ b), w′

1 ≤ w1 − 2(t2 − 1) + g(A(T ) ∩A(S)) + 2(t− a)(t− b) ≤2 − a(t− b) − b(t− a) ≤ w1, and w′

2 < w2.

(iii) If g(v) = 0 and δ(v)+ �= ∅ for some v ∈ V , then let π′v = πv + 1 and

y′uv = yuv −1 for each uv ∈ δ(v)+, and therefore w′1 < w1. If |δ(v)+| ≥ g(v) > 0, then

we do the same operation, but, in this case, w′1 ≤ w1, w

′2 = w2, and w′

3 < w3.

(iv) Otherwise, let λ′v = λv − 1 and y′uv = yuv − 1 for every uv ∈ δ(v)+, and thus

w′1 ≤ w1, w

′2 = w2, w

′3 ≤ w3, and w′

4 < w4.

(v) If e ∈ E(T ) is not tight, then let z′T = zT − 1 and yh = yh + 1 for everyh ∈ E(T ) − {e}, and thus w1 ≤ w′

1 and w′2 < w2. Let C be the set of components

of (V,Etight,0)[V (T )]. By (iii), |δ(v)+| ≤ t− 1, and so C does not contain singletons.Using that c is T -induced, it can be seen that if I ∈ C and ij, jk, kl ∈ E(I), thenil ∈ E(I); i.e., C contains only complete bipartite graphs. If |C| = 1, then we aredone. Otherwise, T has a K2,2-subgraph ({a1, a2, b1, b2}, {a1b1, a1b2, a2b1, a2b2}) s.t.a1b1, a2b2 ∈ Etight,0 and a1b2, a2b1 ∈ Etight,+, which contradicts that c is T -induced.

(vi) By (ii), there is a unique T ∈ T+ s.t. v ∈ T , and we may assume v ∈ A(T ).If g(v) ≤ t− 1, then let z′T = zT − 1 and π′

a = πa for every a ∈ A(T ). Then w′1 ≤ w1

and w′2 < w2.

(vii) Suppose that the statement does not hold for some v ∈ A(T ). Then letλ′v = λv − 1, z′T = zT − 1, and y′ab = yab + 1 for every a ∈ A(T ) − {v}, b ∈ B(T ).

Then w′1 ≤ w1 and w′

2 < w2.

(viii) If uv ∈ E+, u ∈ A(T ), and v ∈ B−B(T ), then let z′T = zT −1, y′uv = yuv−1,and π′

a = πa + 1 for every a ∈ A(T ). Then w′1 ≤ w1 and w′

2 < w2.

(ix) If πv > 0 and λv > 0 for some v ∈ V , then let π′v = πv − 1 and λ′

v = λv − 1.Then w′

1 ≤ w1, w′2 ≤ w2, w

′3 ≤ w3, and w′

4 < w4.

We define a graph G′ from the graph (V,Etight,0) by shrinking A(T ) and B(T )to new vertices TA and TB for each T ∈ T+; we delete

⋃{E(T ) : T ∈ T+} from the

edge set; and finally, to obtain a simple graph, we delete the parallel copies of edges.Thus, the set of old and new vertices in G′ is Vold = V −

⋃{V (T ) : T ∈ T+} and

Vnew = {TA, TB : T ∈ T+}. The sets Aold, Bold, Anew, and Bnew are defined similarly.

In order to construct a T -free (f, g)-factor of G, we try to construct a T ′-free(l, u)-factor of G′ with u : V (G′) → Z, l : V (G′) → Z, and T ′. First, let

u(v) =

⎧⎨⎩

g(v) − |δ(v)+| if v ∈ Vold and λv = 0,f(v) − |δ(v)+| if v ∈ Vold and λv > 0,

1 if v ∈ Vnew,

l(v) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

g(v) − |δ(v)+| if v ∈ Vold and πv > 0,max(f(v) − |δ(v)+|, 0) if v ∈ Vold and πv = 0,

1 if v ∈ Vnew, v = TX ,and πu + λu > 0 for each u ∈ X ∩ V (T ),

0 if v ∈ Vnew, v = TX ,and πu + λv = 0 for some u ∈ X ∩ V (T ).

Next, let T ′ be a family of subgraphs of G′ containing each T ∈ T s.t. V (T )∩⋃{V (S) :

S ∈ T+} = ∅ and G′[V (T )] is isomorphic to Kt,t.

Case 1. G′ has a T ′-free (l, u)-factor H ′. For satisfying the complementary

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354 MARTON MAKAI

slackness conditions with respect to (y, π, λ, z), we have to define H s.t.

E(H) ⊆ Etight,

ye > 0 ⇒ e ∈ E(H),

πv > 0 ⇒ |δH(v)| = g(v),

λv > 0 ⇒ |δH(v)| = f(v),

zT > 0 ⇒ |E(H[T ])| = t2 − 1.

First, each edge of H ′ has an inverse image at the shrinking operation. Thus, foreach e ∈ E(H ′), we put exactly one of these edges into H. Next, for T ∈ T+, A(T ) isincident with at most one edge of (the already defined) H. If there is such an edge,then let aT be its end vertex in A(T ). Otherwise, πu+λu = 0 for some u ∈ A(T ), andthus we let aT = u. We choose bT similarly. For each T ∈ T+, we put E(T )−{aT bT }into H. Last, we put E+ into H.

The construction shows that H is an (f, g)-factor of G and that it meets thecomplementary slackness conditions. We have to prove only that H is T -free. Clearly,for T ∈ T+, H does not contain T as a subgraph. Suppose now that H contains T forsome T ∈ T0 and T shares some vertices with an S ∈ T+. Now dH(A(S), B−V (S)) ≤ 1and dH(B(S), A−V (S)) ≤ 1, and thus |B(T )−B(S)| ≤ 1 and |B(T )−B(S)| ≤ 1. As Tand S are different, we may assume |A(T )−A(S)| = 1. Thus, either |B(T )−B(S)| = 0,or |B(T )−B(S)| = 1. It is not hard to see that both lead to contradiction. So considerT ∈ T with V (T ) ⊆ V −

⋃{V (S) : S ∈ T+}, and suppose that T is a subgraph of

H. Let C be the set of components of the graph (V,Etight,0)[V (T )]. Using that cis T -induced, it can be seen that if I ∈ C and ij, jk, kl ∈ E(I), then il ∈ E(I);i.e., C contains only complete bipartite graphs. If |C| = 1, then E(T ) ⊆ Etight,0, Tis forbidden in H ′, and H ′ cannot contain T , which is a contradiction. As E(T ) ⊆Etight,0 ∪ Etight,+, (iii) implies that C does not contain singletons. Hence, |C| ≥ 2,and T has a K2,2-subgraph ({a1, a2, b1, b2}, {a1b1, a1b2, a2b1, a2b2}) s.t. a1b1, a2b2 ∈Etight,0 and a1b2, a2b1 ∈ Etight,+, which contradicts that c is T -induced.

Case 2. G′ has no T ′-free (l, u)-factor. In this case, we construct the dual solution(y′, π′, λ′, z′) so that

∑e∈E

y′e +∑v∈V

(π′vg(v) − λ′

vf(v)) +∑T∈T

(t2 − 1)z′T

<∑e∈E

ye +∑v∈V

(πvg(v) − λvf(v)) +∑T∈T

(t2 − 1)zT .

By Theorem 6, there exists X ′ ⊆ A′(G′), Y ′ ⊆ B′(G′) satisfying

l(X ′) > u(Y ′) + iG′(X ′ ∪B′(G′) − Y ′) − cT ′(G′[X ′ ∪B′(G′) − Y ′])(2.2)

or

l(Y ′) > u(X ′) + iG′(Y ′ ∪A′(G′) −X ′) − cT ′(G′[Y ′ ∪A′(G′) −X ′]).(2.3)

By symmetry, we may assume that (2.2) holds. Moreover, we choose X ′ and Y ′ so thatX ′∪B′(G′)−Y ′ is minimal. Let C′ be the set of T ′-components of G′[X ′∪B′(G′)−Y ′].Let I ′ be the set of edges of E(G′[X ′∪B′(G′)−Y ′]) which are not in T ′-components,and let I ′T ′ be the set of edges of E(G′[X ′ ∪B′(G′) − Y ′]) in T ′-components.

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Kt,t-FREE t-MATCHINGS OF BIPARTITE GRAPHS 355

Lemma 8. Let x ∈ X ′. If x ∈ V (T ) for some T ′-component T of G′[X ′∪B′(G′)−Y ′], then l(x) = t, and |I ′ ∩ δG′(x)| < l(x) otherwise.

Let y ∈ B′(G′)−Y ′. If y ∈ V (T ) for some T ′-component T of G′[X ′∪B′(G′)−Y ′],then u(y) = t, and |I ′ ∩ δG′(y)| < u(y) otherwise.

Proof. Otherwise, for the first case, we reset X ′ to X ′ − x, and we reset Y ′ toY ′ + y for the second one.

If x ∈ X ′, y ∈ B′(G′)−Y ′, and xy ∈ I ′, then, by Lemma 8, l(x) ≥ 2 and u(y) ≥ 2,and hence x and y cannot be shrunk vertices. This implies that each edge of I ′ has aunique inverse image at the shrinking operation. Let X and Y be the inverse imagesof X ′ and Y ′ at the shrinking operation.

The dual solution (y, π, λ, z) changes as follows. Let

y′ab =

⎧⎪⎪⎨⎪⎪⎩

yab − 1 if a ∈ A−X and b ∈ Y and yab > 0,yab + 1 if a ∈ X and b ∈ B − Y and yab > 0,

yab + 1 = 1 if ab ∈ I ′,yab otherwise,

π′v =

⎧⎨⎩

πv − 1 if v ∈ X and πv > 0,πv + 1 if v ∈ Y and λv = 0,

πv otherwise,

λ′v =

⎧⎨⎩

λv + 1 if v ∈ X and πv = 0,λv − 1 if v ∈ Y and λv > 0,

λv otherwise,

z′T =

⎧⎪⎪⎨⎪⎪⎩

zT − 1 if TA ∈ A′(G′) −X ′, TB ∈ Y ′, and zT > 0,zT + 1 if TA ∈ X ′, TB ∈ B′(G′) − Y ′, and zT > 0,

zT + 1 = 1 if T ∈ I ′T ′ ,zT otherwise.

First, it easily follows from the construction that (y′, π′, λ′, z′) ≥ 0 and the dualinequality (2.1) remains true for (y′, π′, λ′, z′).

Next, we have to compute the change of the dual objective function. If x ∈ X ′ isa shrunk vertex, then let x1, x2, . . . , xt be the inverse vertices. Then either πxi > 0and g(xi) = t, or, if πxi

= 0, then l(x) > 0 implies l(x) = 1, πxi+ λxi

> 0, andtherefore f(xi) = t by (vii). Also, δ(xi)+ = ∅. Next, suppose that x ∈ X ′ is notshrunk, but x is in some T ′-component of G′[X ′ ∪ (B′(G′) − Y ′)]. If πx > 0, thent = l(x) = g(x) − |δ(x)+| by Lemma 8, and if πx = 0, then t = l(x) = f(x) − |δ(x)+|also by Lemma 8. Last, suppose that x ∈ X ′ is not shrunk, but x is not in someT ′-component of G′[X ′ ∪ (B′(G′)− Y ′)]. If πx > 0, then l(x) = g(x)− |δ(x)+|, and ifπx = 0, then l(x) = f(x) − |δ(x)+|. These together imply

∑x∈X

d+(v) −∑

v∈X,πv>0

g(v) −∑

v∈X,πv=0

f(v) + (t2 − 1)|T+[X ∪B]| = −l(X ′).

Similarly, let y ∈ Y ′. If y is shrunk and y1, y2, . . . , yt are the inverse vertices, theneither λyi

= 0 and g(yi) = t, or, if λyi> 0, then f(yi) = t by (vii). If y is not shrunk,

then either λy = 0 and u(y) = g(y) − |δ(y)+|, or λy = 0 and u(y) = f(y) − |δ(y)+|.Thus,

−∑x∈Y

d+(v) +∑

v∈Y,λv=0

g(v) +∑

v∈Y,λv>0

f(v) − (t2 − 1)|T+[A ∪ Y ]| = u(Y ′).

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356 MARTON MAKAI

The change of the dual objective function is

∑e∈E

(y′e − ye) + g(v)∑v∈V

(π′v − πv) + f(v)

∑v∈V

(λ′v − λv) + (t2 − 1)

∑T∈T

(z′T − zT )

= |E+[X ∪ (B − Y )]| − |E−[(A−X) ∪ Y ]| + |I ′|

−∑

v∈X,πv>0

g(v) −∑

v∈X,πv=0

f(v) +∑

v∈Y,λv=0

g(v) +∑

v∈Y,λv>0

f(v)

+ (t2 − 1) (|T+[X ∪ (B − Y )]| − |T+[(A−X) ∪ Y ]| + cT ′(G′[X ′ ∪ (B′(G′) − Y ′)]))

=∑v∈X

d+(v) −∑

v∈X,πv>0

g(v) −∑

v∈X,πv=0

f(v) + (t2 − 1)|T+[X ∪B]|

−∑v∈Y

d+(v) +∑

v∈Y,λv=0

g(v) +∑

v∈Y,λv>0

f(v) − (t2 − 1)|T+[A ∪ Y ]|

+|I ′| + (t2 − 1)cT ′(G′[X ′ ∪ (B′(G′) − Y ′)])

= −l(X ′) + u(Y ′) + |I ′| + (t2 − 1)cT ′(G′[X ′ ∪ (B′(G′) − Y ′)]) < 0.

This is a contradiction, finishing the proof.

3. Proof of Theorem 6. The proof of Theorem 6 follows the structure ofFrank’s proof for Theorem 3 [2]. However, it can easily be seen that if an arbi-trary family of Kt,t’s is forbidden, then the argument described there does not work.To address this problem, a slight extension of the Frank–Jordan theorem is used.

Consider now a bipartite graph on the vertex set V = A ∪ B with edge setE = A×B, and let P = {(X,Y ) : ∅ � X ⊆ A, ∅ � Y ⊆ B} be called the set of pairs.For U ⊆ V , P[U ] = {(X1, X2) ∈ P : X1 ∪ X2 ⊆ U}. A pair (X1, X2) is said to betrivial if at least one of X1 and X2 is singleton.

In this section, when we use the word collection, this means a multiset of pairs;i.e., a pair belongs to the collection with multiplicity. Thus, it is rather convenient toconsider a collection of pairs as a nonnegative function mapping P into Z. The sumof two collections is defined by the sum of these functions. Other algebraic operationsare handled similarly.

Two pairs X,Y ∈ P are independent if δE(X) ∩ δE(Y ) = ∅, while a collection ofpairs is called independent if its members are pairwise independent. More generally, acollection F of pairs satisfying

∑U∈F δE(U) ≤ h is called h-independent. (Remember

that sets and their characteristic functions are not distinguished.) We define thepartial order on P as (X1, X2) (Y1, Y2) if and only if X1 ⊆ Y1 and X2 ⊇ Y2. Twopairs X,Y ∈ P are comparable if X Y or Y X. Two pairs are crossing if theyare neither comparable nor independent. A collection of pairs is called cross-free if itcontains no two crossing pairs.

If we are given a function p : P → Z, we say that the pair X is positive ifp(X) > 0. The nonnegative function p : P → Z is said to be skew-bisupermodular if,for every two positive crossing pairs X = (X1, X2) and Y = (Y1, Y2), there exists across-free collection of positive pairs GX,Y satisfying

(3.1) δE(X) + δE(Y ) ≥∑

U∈GX,Y

δE(U) and p(X) + p(Y ) ≤∑

U∈GX,Y

p(U)

and, for any collection H of positive pairs and any sequence of collections

(3.2) H = H0,H1,H2, . . . ,

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Kt,t-FREE t-MATCHINGS OF BIPARTITE GRAPHS 357

where Hi+1 is obtained from Hi by decreasing the multiplicities of two crossing pairsX,Y ∈ Hi by 1 and increasing the multiplicities of the members of GX,Y by 1, resultingin a finite sequence. Hence, the last member UH is a cross-free collection.

The nonnegative function z : E → Z is said to be a cover of p if p(X1, X2) ≤z(δE(X1, X2)) for every (X1, X2) ∈ P. The cardinality of a minimum cover of p isτp = min

∑e∈E z(e), where the minimum ranges over all covers z of p. Similarly,

max∑

U∈F p(U) is denoted by νp, where the maximum is taken over all independentcollections F .

Theorem 9. For any skew-bisupermodular function p, νp = τp.Proof. νp ≤ τp can be seen easily. If νp = 0, then z = 0 is a cover, and F = ∅ is an

independent collection, which together give equality. Thus, νp > 0 can be assumed.Similarly, we assume |A||B| ≥ 2. We can observe that, for any e ∈ E , the functionpe : P → Z defined by pe(X) = max{p(X) − d{e}(X), 0} is a skew-bisupermodularfunction. If there exists an edge e ∈ E s.t. νpe ≤ νp−1, then τp ≤ τpe +1 = νpe +1 ≤ νpby using τp ≤ τpe + 1 by induction, and we are done. Thus, νpe = νp for every e ∈ E .For each e ∈ E , let us consider the independent collection He s.t. p(He) = νpe , andlet H =

∑e∈E He. By the construction, p(H) = νp|A||B|, and H is |A||B| − 1-

independent. According to the definition of skew-bisupermodularity, there exists across-free collection UH which is |A||B| − 1-independent, and p(UH) ≥ νp|A||B|. Byapplying Dilworth’ theorem to the partial order restricted to UH, UH decomposesinto at most |A||B| − 1 antichains s.t. each member of UH is contained in as many

antichains as its multiplicity. But then there is an antichain A with p(A) ≥ νp|A||B||A||B|−1 >

νp, contradicting the definition of νp.Let us define pAν : 2A → Z by

pAν (Z) = max{p(G) : G is an independent subcollection of P[Z ∪B]}

and pBν : 2B → Z by

pBν (Z) = max{p(G) : G is an independent subcollection of P[A ∪ Z]}.

The proofs of the following four theorems are the same as the proofs of theanalogous theorems of Frank and Jordan [3].

Theorem 10. Let m : V → Z be a nonnegative function with m(A) = m(B),and let p be a skew-bisupermodular function on P. Then there exists a cover z of ps.t. z(δE(v)) = m(v) for every v ∈ V if and only if m(Z) ≥ pAν (Z) for every Z ⊆ Aand m(Z) ≥ pBν (Z) for every Z ⊆ B.

Proof. The necessity can be easily seen. For the sufficiency, we define p′ : P → Z,p′ ≥ 0 by p′(a,B) = m(a), p′(A, b) = m(b) for a ∈ A, b ∈ B, and p′(X1, X2) =p(X1, X2) for the other pairs. p(a,B) ≤ pAν (a) ≤ m(a), and similarly p(A, b) ≤ m(b).Therefore, p ≥ p′, and p′ is skew-bisupermodular, since it is obtained from p byincreasing the value on the trivial pairs (a,B) and (A, b), and these pairs cross noother. Let z be a minimum cover of p′. Clearly, z(δE(v)) ≥ m(v) for each v ∈ V .If we have equality for each v, then we are done. Thus, there exists an independentcollection F s.t. m(A) = m(B) < z(E) = p′(F). F cannot contain trivial pairs ofform both (a,B) and (A, b), and we may assume that it contains only pairs of thefirst type. Then let Z = {a ∈ A : (a,B) ∈ F} and F ′ = F − {(a,B) : a ∈ A}.Then m(A) < p′(F) implies m(A − Z) < p′(F ′) = p(F ′) ≤ pAν (A − Z), which is acontradiction.

The following statement can be proved similarly.

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358 MARTON MAKAI

Theorem 11. Let m : A → Z be a nonnegative function, and let p be a skew-bisupermodular function on P. Then there exists a cover z of p s.t. z(δE(v)) = m(v)for every v ∈ A if and only if m(Z) ≥ pAν (Z) for every Z ⊆ A.

We call a function q : V → Z supermodular if q(X)+q(Y ) ≤ q(X ∩Y )+q(X ∪Y )holds for every X,Y ⊆ V . If, in addition, q(∅) = 0 and q is monotone increasing(i.e., q(X) ≤ q(Y ) whenever X ⊆ Y ), then q is said to be a contra-polymatroidfunction. The polyhedron C(q) = {x ∈ RV : x(U) ≥ q(U) ∀U ⊆ V } is said to be thecontra-polymatroid defined by q. We will use that contra-polymatroids are integerpolyhedra.

Theorem 12. pAν and pBν are contra-polymatroid functions.

Proof. By symmetry, it is enough to prove the statement for pAν . It is clearthat pAν is nonnegative, monotone increasing and pAν (∅) = 0. Thus, we have to provesupermodularity, i.e., the inequality pAν (X)+pAν (Y ) ≤ pAν (X∩Y )+pAν (X∪Y ) for everyX,Y ⊆ A. Let GX and GY be collections which give the maximum in the definition ofpAν (X) and pAν (Y ). Then we can apply the uncrossing procedure to G = GX+GY whichresults in a cross-free family UG . Clearly, for each ab ∈ E , d{ab}(UG) ≤ 2 if a ∈ X ∩Y ,d{ab}(UG) ≤ 1 if a ∈ X ∪ Y −X ∩ Y , and d{ab}(UG) = 0 if a ∈ A−X ∪ Y . Let Umin

Gconsist of the minimal elements UG with respect to the partial order . (If a minimalelement has multiplicity 2, then it is taken only once.) Then Umin

G is an independentcollection of P[(X∩Y )∪B], and UG−Umin

G is an independent collection of P[X∪Y ∪B].Thus, pAν (X) + pAν (Y ) ≤ p(Umin

G ) + p(UG − UminG ) ≤ pAν (X ∩ Y ) + pAν (X ∪ Y ).

Theorem 13. Let g : V → Z be a nonnegative function, and let p be a skew-bisupermodular function on P. Then there exists a cover z of p s.t. z(δE(v)) ≤ g(v)for every v ∈ V if and only if g(Z) ≥ pAν (Z) for every Z ⊆ A and g(Z) ≥ pBν (Z) forevery Z ⊆ B

Proof. The restriction of g to A is in C(pAν ), and hence there is a minimal memberm : A → Z of C(pAν ) s.t. m(a) ≤ g(a) for each a ∈ A. Similarly, we can considera minimal member m : B → Z of C(pBν ) s.t. m(b) ≤ g(b) for each b ∈ B. Theinteger members of these two contra-polymatroids are the degree sequences of covers,and hence m(A) = m(B). Then there exists a cover with degree function m, whichcompletes the proof.

Proof of Theorem 6. If there exists a T -free (l, u)-factor, then for X ⊆ A andY ⊆ B, (1.9) and (1.10) clearly hold. We prove now the opposite direction, thussupposing that (1.9) and (1.10) hold for every X ⊆ A and Y ⊆ B.

We define a skew-bisupermodular function p : P → Z. For every T ∈ T , letp(A(T ), B(T )) = 1. If a ∈ A, ∅ �= Z ⊆ B, and G[a∪Z] is a complete bipartite graph,then let p(a, Z) = max{|Z| − u(a), 0}. Similarly, if b ∈ B, ∅ �= Z ⊆ A, and G[Z ∪ b]is a complete bipartite graph, then let p(Z, b) = max{|Z| − u(b), 0}. On other pairs,p is defined to be 0.

Lemma 14. p is skew-bisupermodular.

Proof. First, p is nonnegative. Second, suppose that X and Y are positivecrossing pairs. If X = (a,B1) and Y = (a,B2) are trivial pairs, then let GX,Y ={(a,B1 ∩B2), (a,B1 ∪B2)}. If X = (a,B1) is trivial and Y = (T1, T2) is a forbiddenKt,t, then let GX,Y = {(a,B1 ∪ T2)}. If X = (T1, T2) and Y = (S1, S2) are forbiddenKt,t’s, then let GX,Y = {(a, T2 ∪ S2) : a ∈ T1 ∩ S1} ∪ {(T1 ∪ S1, b) : b ∈ T2 ∩ S2}. Itcan easily be checked that (3.1) is satisfied.

Thus, we have to prove the existence of the sequence (3.2) for every collection H.Suppose that H0 = H,H1,H2, . . . ,Hi have already been defined. If Hi is cross-free,then we are done with UH = Hi. Otherwise, Hi contains two crossing pairs X and Y .

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Kt,t-FREE t-MATCHINGS OF BIPARTITE GRAPHS 359

Then Hi+1 is obtained from Hi by decreasing the multiplicities of X and Y and byincreasing the multiplicities of the members of GX,Y by 1. The uncrossing operationfor a trivial pair and a Kt,t or for two Kt,t’s decreases the sum of multiplicities of Kt,t’s,and these multiplicities cannot increase in other operations. Hence, such an uncrossingcan be applied finitely many times. When two crossing trivial pairs are uncrossed,then

∑U∈Hj

(|U1| − |U2|)2 increases, which is upper bounded by |Hj |max{|A|, |B|}2.Hence, the number of Kt,t pairs decreases finitely many times, and between each twosuch operations, there are finitely many other operations.

Next, g : V → Z is defined by g(v) = dG(v) − l(v) for every v ∈ V . By applying(1.9) to X = {v} and Y = ∅, we get that g(v) ≥ 0 for every v ∈ A. Similarly,g(v) ≥ 0 for every v ∈ B by (1.10). Now we ask whether a cover z of p exists s.t.z(δE(v)) ≤ g(v) for every v ∈ V .

Case 1. There exists such a cover. Then let z be a minimal cover satisfyingz(δE(v)) ≤ g(v) for every v ∈ V . (Minimal means that

∑e∈E z(e) is as small as

possible.)

Lemma 15. If z(ab) > 0, then ab ∈ E. Moreover, z is 0 − 1 valued, and{ab ∈ E : z(ab) = 0} is a T -free (l, u)-factor.

Proof. If z(ab) > 0, then there exists a positive pair (X1, X2) s.t. a ∈ X1, b ∈ X2,and z(δE(X1, X2)) = p(X1, X2). But this implies ab ∈ E. If z(ab) ≥ 2, then there isa trivial pair X = (X1, X2), a ∈ X1, b ∈ X2 s.t. z(δE(X)) = p(X). Suppose |X1| = 1.Then z does not cover (X1, X2 − {b}), which is a contradiction.

The fact that {ab ∈ E : z(ab) = 0} is a T -free (l, u)-factor easily follows from thedefinition of p and g.

Then, by Lemma 15, we are done.

Case 2. There does not exist such a cover. Then, by Theorem 13, there exists aset Z ⊆ A s.t. g(Z) < pAν (Z) or Z ⊆ B s.t. g(Z) < pBν (Z). By symmetry, we mayassume the first, and let us choose Z to be minimal among these sets. Let G be afamily which gives the maximum in the definition of pAν (Z). Suppose, moreover, thatthe number of Kt,t pairs in G is as small as possible, and, subject to this, the numberof its trivial pairs is minimal.

Lemma 16. For every a ∈ A, G contains at most one trivial pair of form (a,B1).Similarly, for every b ∈ B, G contains at most one trivial pair of form (A1, b).

Proof. If (a,B1) are (a,B2) trivial pairs in G, then G could be replaced by G −{(a,B1), (a,B2)}+ (a,B1 ∪B2). This operation does not decrease p(G) and does notchange the number of Kt,t pairs but decreases the number of trivial pairs, which is acontradiction.

Lemma 17. If (T1, T2) is a Kt,t member of G, a ∈ T1, then G has no trivialmember of form (a,B1). Similarly, for b ∈ T2, G has no trivial member of form(A1, b).

Proof. Otherwise, G could be replaced by G−{(a,B1), (T1, T2)}+(a,B1∪T2). Thisoperation does not decrease p(G) but decreases the number of Kt,t pairs in G.

Lemma 18. If (X1, X2) and (Y1, Y2) are two Kt,t’s of G, then (X1 ∪X2) ∩ (Y1 ∪Y2) = ∅.

Proof. If (X1∪X2)∩(Y1∪Y2) �= ∅, then, by symmetry, we may suppose X1∩Y1 �= ∅and X2 ∩ Y2 = ∅. Then we could remove (X1, X2) and (Y1, Y2) from G and insert(a,X2 ∪ Y2) into G for every a ∈ X1 ∩ Y1. This operation does not increase p(G) butdecreases the number of Kt,t pairs in G.

Lemma 19. G contains no trivial pair of form (a,B1).

Proof. Suppose (a,B1) ∈ G. By the above lemmas, there is no pair (X1, X2) ∈ G

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360 MARTON MAKAI

s.t. a ∈ X1. Thus, let Z ′ = Z − {a} and G′ = G − {(a,B1)}. By dG(a) ≥ |B1| andl(a) ≤ u(a), dG(a)− l(a) ≥ |B1| −u(a). And, finally, g(Z ′) = g(Z)− (dG(a)− l(a)) <p(G) − (|B1| − u(a)) = p(G′), which contradicts the minimal choice of Z.

Thus, G is composed of the trivial pairs (A1, b1), (A2, b2), . . . , (Ar, br) and of Kt,t

pairs (X1, Y2), (X2, Y2), . . . , (Xs, Ys) s.t. the sets Xi, Yj and the singletons bk arepairwise disjoint. Let Y = {b1, b2, . . . , br} and X = Z. We will show that X and Ycontradict (1.9).

Lemma 20. Let (T1, T2) ∈ G be a Kt,t pair, and let ab ∈ E. If a ∈ T1 andb ∈ B − T2, or a ∈ Z − T1 and b ∈ T2, then there is a pair (X1, X2) ∈ G s.t. a ∈ X1

and b ∈ X2.Proof. Let us prove the a ∈ T1, b ∈ B − T2 case. If the statement does not hold,

then we can replace G by G − {(T1, T2)} + {(a, {b} ∪ T2)}, which would decrease thenumber of Kt,t pairs of G.

These lemmas together imply that if (T1, T2) ∈ G is a Kt,t pair, then there is noedge ab ∈ E s.t. a ∈ Z − T1 and b ∈ T2. The definition of G implies that Ak = Z forevery k = 1, 2, . . . , r.

Then (X1, Y2), (X2, Y2), . . . , (Xs, Ys) define some of the Kt,t components of G[Z∪B − Y ]. Hence,

iG(Z ∪ Y ) − u(Y ) + s = p(G) > g(Z) =∑v∈Z

(dG(v) − l(v)) = iG(Z ∪B) − l(Z),

which completes the proof.

Acknowledgments. The author wishes to thank Andras Frank, Tamas Kiraly,Zoltan Kiraly, and Zsuzsanna Vaik for useful discussions on the topic.

REFERENCES

[1] T. Fleiner, Uncrossing a family of set-pairs, Combinatorica, 21 (2001), pp. 145–150.[2] A. Frank, Restricted t-matchings in bipartite graphs, Discrete Appl. Math., 131 (2003),

pp. 337–346.[3] A. Frank and T. Jordan, Minimal edge-coverings of pairs of sets, J. Combin. Theory Ser.

B, 65 (1995), pp. 73–110.[4] J. F. Geelen, The C6-free 2-Factor Problem in Bipartite Graphs is NP-Complete, manuscript,

1999.[5] D. Hartvigsen, The square-free 2-factor problem in bipartite graphs, in Integer Program-

ming and Combinatorial Optimization (Graz, 1999), Lecture Notes in Comput. Sci. 1610,Springer, Berlin, 1999, pp. 234–241.

[6] D. Hartvigsen, Finding maximum square-free 2-matchings in bipartite graphs, J. Combin.Theory Ser. B, 96 (2006), pp. 693–705.

[7] Z. Kiraly, The minimum cost square-free 2-factor problem in bipartite graphs is NP-complete,private communication.

[8] Z. Kiraly, Kt,t-Free t-Matchings in Bipartite Graphs, manuscript, 2000.[9] Z. Kiraly, C4-Free 2-Factors in Bipartite Graphs, Tech. report TR-2001-13, MTA-ELTE

EGRES, Egervary Research Group on Combinatorial Optimization, Budapest, Hungary,2001.

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