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Fractionally Balanced Hypergraphs and Rainbow KKM Theorems

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Fractionally Balanced Hypergraphs and Rainbow KKM Theorems FRACTIONALLY BALANCED HYPERGRAPHS AND RAINBOW KKM THEOREMS RON AHARONI, ELI BERGER, JOSEPH BRIGGS, EREL SEGAL-HALEVI, AND SHIRA ZERBIB Abstract. A d-partite hypergraph is called fractionally balanced if there ex- ists a non-negative function on its edge set that has constant degrees in each vertex side. Using a topological version of Hall’s theorem we prove lower bounds on the matching number of such hypergraphs. These, in turn, yield results on mulitple-cake division problems and rainbow matchings in families of d-intervals. 1. Introduction 1.1. Multidimensional and rainbow versions of the KKM theorem. Given a polytope P , we denote by V (P ) its set of vertices. The (n − 1)-dimensional Rn simplex ∆n−1 is the set of points ~x = (x1,...,xn) ∈ + satisfying xi = 1. Its vertices are ei, i ≤ n, where ei(j) = δ(i, j), namely ei(i)=1, ei(j)=0 for j 6= i. A well-known continuous version of the even better-known Sperner’sP lemma is: Theorem 1.1 (The KKM theorem [14]). Let Av, v ∈ V (∆n−1), be closed subsets of ∆n−1. If (1.1) σ ⊆ Av v∈σ [ for every face σ of ∆n−1, then v∈V (∆n−1) Av 6= ∅. A collection of closed sets ATv, v ∈ V (∆n−1) satisfying (1.1) is called a KKM- cover. Gale [11] proved a rainbow (i.e., colorful) version, in which there are n KKM- i covers (“colors”) Av, v ∈ V (∆n−1), i ∈ [n], and each contributes a set to the arXiv:2011.01053v1 [math.CO] 2 Nov 2020 intersecting sub-collection: R. Aharoni: Department of Mathematics, Technion, Israel and MIPT, Dolgoprudny, Russia. [email protected]. Ron Aharoni is Supported by the Israel Science Foundation (ISF) grant no. 2023464 and the Discount Bank Chair at the Technion. This paper is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 823748. This work was supported by the Russian Federation Government in the framework of MegaGrant no. 075-15-2019-1926 when Ron Aharoni worked on Sections 6 through 8 of the paper. E. Berger: Department of Mathematics, University of Haifa, Israel. [email protected]. J. Briggs: Department of Mathematics, Technion, Israel. [email protected]. E. Segal-Halevi: Department of Computer Science, Ariel University, Israel. [email protected]. Segal-Halevi is supported by an Israel Science Foundation no. 712/20. S. Zerbib: Department of Mathematics, Iowa State University, USA. [email protected]. S. Zerbib is supported by NSF grant DMS-1953929. R. Aharoni, E. Berger and S. Zerbib are supported by BSF grant no. 2016077. 1 2 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB i i Theorem 1.2 (Rainbow KKM [11]). For every i ∈ [n] let A = Av, v ∈ V (∆n−1) be a KKM-cover of ∆n−1. Then there exists a permutation π of V (∆n−1) such that Ai 6= ∅. i∈[n] π(ei) T (As noted, the vertices of ∆n−1 are ei.) The notion of KKM-covers can be extended to general polytopes. Given a poly- tope Q, a KKM-cover is a collection of closed sets Av defined for every vertex v of Q, satisfying (1.2) σ ⊆ Av v∈[V (σ) for every face σ of Q. Remark 1.3. In more general versions, a set is assigned to every face, not only to vertices. See Theorem 2.5 below. In this paper we will focus on polytopes Q that are products of simplices. We i i shall prove results of the form “given a family of KKM-covers A = (Av | v ∈ V (Q)), there exists a large rainbow subfamily that has non-empty intersection”. We shall pose a further requirement, that the vertices v participating in the rainbow set are disjoint (in an intuitive sense, to be made precise below) - this is necessary for our main application, multiple cakes division. We shall use two main tools: (1) Theorem 2.4. Given a cake-division problem with d cakes, the theorem produces a (d + 1)-partite hypergraph, with weights on the edges, that are evenly distributed in every vertex side. We shall call such hypergraphs fractionally balanced. The theorem then asserts that the number of agents that can be satisfied by some division of the cakes is at least the matching number of this hypergraph. This theorem is inspired by a result of Meunier and Su [19]. (2) A topological version of Hall’s theorem. It will be used to find lower bounds on the matching numbers of fractionally balanced hypergraphs. 1.2. Division of multiple cakes. The bulk of the paper deals with partitions of multiple cakes. A “cake” in this setting is a copy of the unit interval [0, 1]. A partition of [0, 1] into a interval pieces can be identified with the vector (x1,...,xa) of the lengths of the pieces, listed from left to right, and since xi = 1, such a partition can be viewed as an element of ∆a−1. P Given d “cakes” C1,...,Cd, we consider partitions P of their union, the t-th cake 1 1 d d Ct being partitioned into at slices. Then P = ((P1 ,...,Pa1 ),..., (P1 ,...,Pad )) is d an element of P := t=1 ∆at−1. The subintervals of Ct in the partition P are t t denoted by Ij (P ), j ≤ at, when ordered from left to right. So, the length of Ij (P ) t Q is Pj . We denote by J the set of all vectors ~j = (j1,...,jd), jt ∈ [at]. For every vector ~ ~ j = (j1,...,jd) ∈ J , let v(j) = (ej1 ,ej2 ,...ejd ) be the vertex of P corresponding to ~j. Our notation will sometimes not distinguish between ~j and v(~j). There is a set of “agents” (or “players”). Given a partition P , we wish to allocate to each agent a d-tuple of slices, one from each cake. Such a d-tuple is determined ~ ~ t by a vertex v(j) of P (that is, by a vector j ∈ J ) — choosing the slice Ijt from Ct FRACTIONALLY BALANCED HYPERGRAPHS 3 for each t ∈ [d]. Of course, we want the d-tuples ~ja and ~jb of slices allocated to two different agents a,b to be disjoint, namely component-wise distinct. The agents are choosy. Each agent i has, for each partition P ∈ P, a list Li(P ) of acceptable d-tuples ~j of slices, indicated by their indices. So, for example, i L (P )= {(3, 5, 2), (1, 4, 2)} (here d =2, a1 = a2 = 3) means that agent i is ready 1 2 3 1 2 3 i to accept in the partition P either (I3 , I5 , I2 ) or (I1 , I4 , I2 ). Thus, L is a multi- valued function from P to J . Its inverse is denoted by Ai. Formally, For every vector ~j ∈ J and every agent i, i i −1 ~ ~ i A~j := (L ) (j)= {P ∈ P | j ∈ L (P )}. So Ai is the set of partitions in which agent i is ready to accept the d-tuple ~j. ~j The next observation expresses natural conditions on the lists Li(P ) as a KKM- cover condition on the sets Ai . It is folklore among cake-dividers, but we have not ~j found an explicit formulation of it that we can cite. The support supp(P ) of a point P ∈P is the minimal face of P containing P . Observation 1.4. For every i, the sets Ai , ~j ∈ V (P), form a KKM-cover of P if ~j and only if (1) they are closed, and (2) for every partition P ∈ J there exists ~j ∈ V (P) such that P ∈ Ai and ~j t Pjt > 0 for every t ∈ [d]. Proof. Suppose that (1) and (2) hold. Let σ be a face of P and let P ∈ σ. By (2), there exists ~j with P t > 0 for all t ∈ [d] and such that P ∈ Ai . Clearly, then, jt ~j ~ i v(j) is a vertex of P and hence of σ, showing P ∈ v∈V (σ) Av. Thus the collection (Ai , v ∈ V (P)) forms a KKM-cover of P. v S Conversely, suppose that the sets Ai form a KKM-cover of P. Let P ∈P and let ~j S = supp(P ). By (1.2), there exists a vertex v = v(~j) of S, where ~j = (j1,...,jd), with P ∈ Ai . Since v is a vertex in S, P t > 0 for all t ∈ [d], as required in (2). ~j jt Condition (2) is called the “hungry agents” assumption. We shall always assume it, as well as (1). Definition 1.5. A set Q of agents is called placatable if there exists a function q φ : Q → V (P) such that q∈Q Aφ(q) 6= ∅ and the supports supp(φ(q)) are disjoint. The condition means thatT it is possible to placate every player q ∈ Q. A partition q P ∈ q∈Q Aφ(q) yields a division of the cakes, in which if every q ∈ Q receives the slices defined by the vertex φ(q) then she is happy, since φ(q) ∈ Lq. We call such T an allocation “admissible”. Our aim is to prove the existence of large placatable sets. Given a partition P ∈P, the maximal size of a set of agents placatable by P is denoted by νD(P ). Remark 1.6. In the literature the lists Li are called preference lists, and then an admissible division is called envy-free or fair.
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