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TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
On Linear and Semidefinite Programming Relaxations for Hypergraph Matching
"... The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a wellstudied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. I ..."
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The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a wellstudied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. In this paper we analyze different linear and semidefinite programming relaxations for the hypergraph matching problem, and study their connections to the local search method. Our main results are the following: • We consider the standard linear programming relaxation of the problem. We provide an algorithmic proof of a result of Füredi, Kahn and Seymour, showing that the integrality gap is exactly k − 1 + 1/k for kuniform hypergraphs, and is exactly k − 1 for kpartite hypergraphs. This yields an improved approximation algorithm for the weighted 3dimensional matching problem. Our algorithm combines the use of the iterative rounding method and the fractional local ratio method, showing a new way to round linear programming solutions for packing problems. • We study the strengthening of the standard LP relaxation by local constraints. We show that, even after linear number of rounds of the SheraliAdams liftandproject procedure on the standard LP relaxation, there are kuniform hypergraphs with integrality gap at least k − 2. On the other hand, we prove that for every constant k, there is a strengthening of the standard LP relaxation by only a polynomial number of constraints, with integrality gap at most (k+1)/2 for kuniform hypergraphs. The construction uses a result in extremal combinatorics. • We consider the standard semidefinite programming relaxation of the problem. We prove that the Lovász ϑfunction provides an SDP relaxation with integrality gap at most (k + 1)/2. The proof gives an indirect way (not by a rounding algorithm) to bound the ratio between any local optimal solution and any optimal SDP solution. This shows a new connection between local search and linear and semidefinite programming relaxations. 1
Improving integrality gaps via ChvátalGomory rounding
 In Approximation, randomization, and combinatorial optimization
, 2010
"... In this work, we study the strength of the ChvátalGomory cut generating procedure for several hard optimization problems. For hypergraph matching on kuniform hypergraphs, we show that using ChvátalGomory cuts of low rank can reduce the integrality gap significantly even though SheraliAdams rel ..."
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In this work, we study the strength of the ChvátalGomory cut generating procedure for several hard optimization problems. For hypergraph matching on kuniform hypergraphs, we show that using ChvátalGomory cuts of low rank can reduce the integrality gap significantly even though SheraliAdams relaxation has a large gap even after linear number of rounds. On the other hand, we show that for other problems such as kCSP, unique label cover, maximum cut, and vertex cover, the integrality gap remains large even after adding all ChvátalGomory cuts of large rank. 1
Intersecting families are essentially contained in juntas
, 2007
"... A family J of subsets of {1,..., n} is called a jjunta if there exists J ⊆ {1,..., n}, with J  = j, such that the membership of a set S in J depends only on S ∩ J. In this paper we provide a simple description of intersecting families of sets. Let n and k be positive integers with k < n/2, an ..."
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A family J of subsets of {1,..., n} is called a jjunta if there exists J ⊆ {1,..., n}, with J  = j, such that the membership of a set S in J depends only on S ∩ J. In this paper we provide a simple description of intersecting families of sets. Let n and k be positive integers with k < n/2, and let A be a family of pairwise intersecting subsets of {1,..., n}, all of size k. We show that such a family is essentially contained in a jjunta J where j does not depend on n but only on the ratio k/n and on the interpretation of “essentially”. When k = o(n) we prove that every intersecting family of ksets is almost contained in a dictatorship, a 1junta (which by the ErdősKoRado theorem is a maximal intersecting family): for any such intersecting family A there exists an � element i ∈ {1,..., n} such that the number k (which is approximately of sets in A that do not contain i is of order � n−2 k−2 n−k times the size of a maximal intersecting family). Our methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.
The number of kintersections of an intersecting family of rsets
 J. Combin. Theory Ser. A
"... The ErdősKoRado theorem tells us how large an intersecting family of rsets from an nset can be, while results due to Lovász and Tuza give bounds on the number of singletons that can occur as pairwise intersections of sets from such a family. We consider a natural common generalization of these p ..."
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The ErdősKoRado theorem tells us how large an intersecting family of rsets from an nset can be, while results due to Lovász and Tuza give bounds on the number of singletons that can occur as pairwise intersections of sets from such a family. We consider a natural common generalization of these problems. Given an intersecting family of rsets from an nset and 1≤k≤r, how many ksets can occur as pairwise intersections of sets from the family? For k = r and k = 1 this reduces to the problems described above. We answer this question exactly for all values of k and r, when n is sufficiently large. We also characterize the extremal families. 1
Graphs and Combinatorics © SpringerVerlag 1986 Decomposition of the Complete rGraph into Complete rPartite rGraphs*
"... Abstract. For n> r> 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete rgraph on n vertices into q complete rpartite rgraphs. Graham and Pollak showed that fz(n) = n 1. Here we observe that f3(n) = n 2 and show that for every fixed ..."
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Abstract. For n> r> 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete rgraph on n vertices into q complete rpartite rgraphs. Graham and Pollak showed that fz(n) = n 1. Here we observe that f3(n) = n 2 and show that for every fixed r> 2, there are positive constants cx(r) and c2(r) such that q(r) < f,(n) " nf'/2J < c2(r) for all n> r. This solves a problem of Aharoni and Linial. The proof uses some simple ideas of linear algebra. 1.