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A Hall Condition for Normal Hypergraphs Takustraße 7 Konrad-Zuse-Zentrum D-14195 Berlin-Dahlem fur¨ Informationstechnik Berlin Germany ISABEL BECKENBACH,RALF BORNDORFER¨ A Hall Condition for Normal Hypergraphs ZIB Report 15-45 (September 2015) Herausgegeben vom Konrad-Zuse-Zentrum f¨urInformationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Telefon: 030-84185-0 Telefax: 030-84185-125 e-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 A Hall Condition for Normal Hypergraphs Isabel Beckenbach∗ y Ralf Bornd¨orfer∗ September 24, 2015 Conforti, Cornu´ejols,Kapoor, and Vu˘skovi´cgave a Hall-type condition for the non-existence of a perfect matching in a balanced hypergraph. We gen- eralize this result to the class of normal hypergraphs. 1 Introduction Hall's theorem gives a necessary and sufficient condition for the existence of a system of distinct representatives of a family of finite sets. It is equivalent to the following result on matchings in bipartite graphs. Theorem 1.1. [Hal35] A bipartite graph has a perfect matching if and only if for all stable sets S the set N(S) of its neighbors is as least as big as S. This result has been generalized to balanced hypergraphs by Conforti, Cor- nu´ejols,Kapoor, and Vu˘skovi´c[CCKV96] using a linear programming argu- ment. Later, Huck and Triesch [HT02] gave the first combinatorial proof, Schrijver provided the probably shortest proof (Corollary 83.1d in [Sch03]), and recently Scheidweiler [Sch11] gave an alternative one based on an elegant Gallai-Edmonds decomposition. Theorem 1.2. [CCKV96] A balanced hypergraph has no perfect matching if and only if there exists a pair (R; B) of disjoint node sets such that je \ Rj ≥ je \ Bj for all hyperedges e but jRj < jBj. The optimization version of Hall's theorem is K}onig'stheorem which holds for bipartite graphs, balanced hypergraphs, and for the larger class of nor- mal hypergraphs. This brings up the question whether Theorem 1.2 can be generalized to normal hypergraphs. However, the theorem's condition is only necessary but not sufficient, as the following example shows. ∗Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany; Email: [email protected], [email protected] yThe work of this author was supported by BMBF Research Campus MODAL - RailLab 1 1 1 4 4 4' 3 2 3 2 (a) A normal hypergraph without (b) The hypergraph depicted in (a) perfect matching which has no after duplicating vertex 4 has a pair (R; B) violating Hall's con- pair (R; B) violating Hall's con- dition. dition. Figure 1: Does Hall's condition hold for normal hypergraphs? Example 1.1. Take H = (V; E) with V = f1; 2; 3; 4g and E = ff1; 2; 4g, f2; 3; 4g, f1; 3; 4gg, see Figure 1a. This is the smallest normal hypergraph that is not balanced. H has no perfect matching and jRj ≥ jBj for every pair of disjoint node sets R; B with je \ Rj ≥ je \ Bj for all e 2 E. We can fix Hall's condition in Example 1.1 by considering the hypergraph H0 = (V 0;E0) obtained from H by multiplying vertex 4 by 2, that is, by setting V 0 = f1; 2; 3; 4; 40g and E0 = ff1; 2; 4; 40g; f2; 3; 4; 40g; f1; 3; 4; 40gg. Now, B := f1; 2; 3g and R := f4; 40g is a pair violating Hall's condition, see Figure 1b. In Section 2 we show how this idea can be generalized in order to derive a Hall condition for normal hypergraphs, including an analysis of bounds for the multiplicity of a node. We also investigate a deficiency version. Section 3 discusses relations and differences between K}onig'sand Hall's theorem in the graph and hypergraph case. 2 A Hall Condition for Normal Hypergraphs We say that a hypergraph satisfies Hall's condition if jRj ≥ jBj for all disjoint node sets R; B with je \ Rj ≥ je \ Bj for all hyperedges e. Otherwise, H violates Hall's condition for hypergraphs. Analogous to graphs [LP86], we define the deficiency of a hypergraph def(H) := minfjV j − jV (M)j : M is a matching of Hg 2 1 2 3 4 5 6 7 Figure 2: A balanced hypergraph with def(H) = 3 > 1 = d(H) to be the minimum number of vertices that cannot be covered by a matching. In the same vein, we call the maximum violation of Hall's condition in a hypergraph its critical difference d(H) := maxfjBj−jRj : R; B ⊆ V (H);R\B = ;; je\Rj ≥ je\Bj8e 2 E(H)g; and a pair R; B of nodes attaining the maximum at the right hand side of this definition a critical pair; this generalizes the definition of the critical difference in graphs [LM12]. A hypergraph has a perfect matching if and only if its deficiency is zero, and it satisfies Hall's condition if and only if its critical difference is zero. Thus, the result of Conforti, Cornu´ejols,Kapoor, and Vu˘skovi´c[CCKV96] can be restated as def(H) = 0 , d(H) = 0 for all balanced hypergraphs H. For bipartite graphs, and more generally K}onig-Egerv´arygraphs (for a char- acterization of K}onig-Egerv´arygraphs see [Ste79], [Dem79], or [KNP06]), it is known that the deficiency is equal to the critical difference, see for example [LM12]. In hypergraphs, the critical difference gives a lower bound on the deficiency. Indeed, if M is a matching covering as many vertices as possible in a hyper- graph H, and R; B is a critical pair of H, then X X d(H) = jBj − jRj = ( je \ Bj + jB n V (M)j) − ( je \ Rj + jR n V (M)j) e2M e2M ≤ jB n V (M)j − jR n V (M)j ≤ jB n V (M)j ≤ jV n V (M)j = def(H): Huck and Triesch [HT02] observed that the gap between the critical difference and the deficiency can be arbitrarily large even for very simple balanced hypergraphs. Example 2.1. [HT02] Let H by the hypergraph with vertices 1; 2;:::; 2n+1 and hyperedges e1 = f1; : : : ; n + 1g; e2 = fn + 1;:::; 2n + 1g, see Figure 2 for an example with n = 3. As H has only two hyperedges it is obviously balanced. Furthermore, every non-empty matching misses n vertices, so def(H) = n. However, d(H) = 1 as there is no pair R; B ⊆ V with jei \Bj ≤ jei \Rj for i = 1; 2 and jBj−jRj > 1, and thus R = fn + 1g, B = fn; n + 2g is a critical pair. If we could take n copies of the vertex n + 1 into the set R, and all other vertices into B, then we would get a pair R; B with jei \ Rj = n = jei \ Bj (i = 1; 2) and jBj − jRj = 2n − n = n. This means that the deficiency of 3 H equals the critical difference of the hypergraph in which node n + 1 is \multiplied" n times. The multiplication trick of Example 2.1 can be formalized and generalized to derive a Hall condition for normal hypergraphs. To this purpose, we use the vertex multiplication definition according to Berge. Definition 2.1. [Ber89] Let H = (V; E) be a hypergraph, v 2 V a fixed vertex, and λ 2 N. The hypergraph obtained by multiplying v by λ is the hypergraph that arises from H by replacing the vertex v by λ new vertices (v; 1);:::; (v; λ) and every hyperedge e containing v by the new hyperedge e n fvg [ f(v; 1);:::; (v; λ)g. V (c) For c 2 N , H is the hypergraph obtained from H by multiplying each (c) (c) (c) vertex v by cv. We denote by V the set of vertices of H , by E the set of hyperedges of H(c), and for every e 2 E we denote by e(c) the corresponding (c) hyperedge in E . If all entries of c are equal to some constant k 2 N, we also write H(k);V (k);E(k); and e(k). Berge observed that H(c) is balanced if H is balanced. Similarly, if H is normal, then H(c) is also normal. So multiplying the vertices of a hypergraph does not destroy normality. One problem arises when looking at the critical difference of the multiplied hypergraph H(k). Namely, if R; B ⊆ V (k) is a critical pair of H(k) with d(H(k)) = jBj − jRj > 0, then we can define a pair of disjoint vertex sets R0;B0 ⊆ V (lk) by taking l times the number of copies of v 2 V that R or B contains. It holds that je\R0j = l·je\Rj ≥ l·je\Bj = je\B0j for all e 2 E(lk) and jB0j − jR0j = l · (jBj − jRj), and thus d(H(lk)) ≥ l · d(H(k)). This means, that d(H(k)) is unbounded for k ! 1. Which is a problem as we do not know a priori how often we have to multiply the vertices. We can overcome this problem by considering a restricted multiplied critical difference defined as d∗(H(k)) := maxfjBj − jRj : R; B ⊆ V (k);R \ B = ;; je \ Rj ≥ je \ Bj 8e 2 E(k); jB \ f(v; 1); (v; 2);:::; (v; k)gj ≤ 1 8v 2 V g; where the set R is arbitrary but B is only allowed to contain at most one copy of each multiplied node. In particular, we have that d∗(Hk) is bounded for k ! 1. This, together width d∗(Hk) ≥ d∗(Hk0 ) for k ≥ k0, implies that d∗(H(k)) becomes constant, ∗ (k) ∗ (N) i.e., there exists an N 2 N with d (H ) = d (H ) for all k ≥ N.
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