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On the Inapproximability of Vertex Cover on k-Partite k-Uniform Hypergraphs

by Venkatesan Guruswami, Rishi Saket
"... Computing a minimum vertex cover in graphs and hypergraphs is a well-studied optimizaton problem. While intractable in general, it is well known that on bipartite graphs, vertex cover is polynomial time solvable. In this work, we study the natural extension of bipartite vertex cover to hypergraphs, ..."
Abstract - Cited by 6 (0 self) - Add to MetaCart
, namely finding a small vertex cover in k-uniform k-partite hypergraphs, when the k-partition is given as input. For this problem Lovász [16] gave a k 2 factor LP rounding based approximation, and a matching ( k 2 − o(1)) integrality gap instance was constructed by Aharoni et al. [1]. We prove

Rainbow Matchings and Hamilton Cycles in Random Graphs

by Deepak Bal, Alan Frieze , 2014
"... Let HPn,m,k be drawn uniformly from all k-uniform, k-partite hypergraphs where each part of the partition is a disjoint copy of [n]. We let HP (κ) n,m,k be an edge colored version, where we color each edge randomly from one of κ colors. We show that if κ = n and m = Kn log n where K is sufficiently ..."
Abstract - Cited by 1 (0 self) - Add to MetaCart
Let HPn,m,k be drawn uniformly from all k-uniform, k-partite hypergraphs where each part of the partition is a disjoint copy of [n]. We let HP (κ) n,m,k be an edge colored version, where we color each edge randomly from one of κ colors. We show that if κ = n and m = Kn log n where K is sufficiently

Rainbow Hamilton cycles in uniform hypergraphs

by Andrzej Dudek, Alan Frieze, Andrzej Ruciński
"... be the complete k-uniform hypergraph, k ≥ 3, and let ℓ be an integer such that 1 ≤ ℓ ≤ k − 1 and k − ℓ divides n. An ℓ-overlapping Hamilton cycle in K (k) n is a spanning subhypergraph C of K (k) n with n/(k − ℓ) edges and such that for some cyclic ordering of the vertices each edge of C consists of ..."
Abstract - Cited by 6 (1 self) - Add to MetaCart
be the complete k-uniform hypergraph, k ≥ 3, and let ℓ be an integer such that 1 ≤ ℓ ≤ k − 1 and k − ℓ divides n. An ℓ-overlapping Hamilton cycle in K (k) n is a spanning subhypergraph C of K (k) n with n/(k − ℓ) edges and such that for some cyclic ordering of the vertices each edge of C consists

Independence and Matchings in σ-hypergraphs

by Yair Caro, Josef Lauri, Christina Zarb
"... Let σ be a partition of the positive integer r. A σ-hypergraph H = H(n, r, q|σ) is an r-uniform hypergraph on nq vertices which are parti-tioned into n classes V1, V2,..., Vn each containing q vertices. An r-subset K of vertices is an edge of the hypergraph if the partition of r formed by the non-ze ..."
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Let σ be a partition of the positive integer r. A σ-hypergraph H = H(n, r, q|σ) is an r-uniform hypergraph on nq vertices which are parti-tioned into n classes V1, V2,..., Vn each containing q vertices. An r-subset K of vertices is an edge of the hypergraph if the partition of r formed by the non

Coverings and Matchings in r-Partite Hypergraphs

by Douglas S. Altner, J. Paul Brooks , 2010
"... Ryser’s conjecture postulates that, for r-partite hypergraphs, τ ≤ (r − 1)ν where τ is the covering number of the hypergraph and ν is the matching number. Although this conjecture has been open since the 1960’s, researchers have resolved it for special cases such as for intersecting hypergraphs wher ..."
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Ryser’s conjecture postulates that, for r-partite hypergraphs, τ ≤ (r − 1)ν where τ is the covering number of the hypergraph and ν is the matching number. Although this conjecture has been open since the 1960’s, researchers have resolved it for special cases such as for intersecting hypergraphs

Rainbow matchings in r-partite r-graphs

by Ron Aharoni, Eli Berger
"... Given a collection of matchings M = (M1,M2,...,Mq) (repetitions allowed), a matching M contained in ⋃ M is said to be s-rainbow for M if it contains representatives from s matchings Mi (where each edge is allowed to represent just one Mi). Formally, this means that there is a function φ: M → [q] suc ..."
Abstract - Cited by 2 (0 self) - Add to MetaCart
] such that e ∈ M φ(e) for all e ∈ M, and |Im(φ) | � s. Let f(r,s,t) be the maximal k for which there exists a set of k matchings of size t in some r-partite hypergraph, such that there is no s-rainbow matching of size t. We prove that f(r,s,t) � 2 r−1 (s − 1), make the conjecture that equality holds for all

On Linear and Semidefinite Programming Relaxations for Hypergraph Matching

by Yuk Hei Chan, Lap Chi Lau
"... The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a well-studied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. I ..."
Abstract - Cited by 16 (0 self) - Add to MetaCart
proof of a result of Füredi, Kahn and Seymour, showing that the integrality gap is exactly k − 1 + 1/k for k-uniform hypergraphs, and is exactly k − 1 for k-partite hypergraphs. This yields an improved approximation algorithm for the weighted 3-dimensional matching problem. Our algorithm combines

COMPUTING THE PARTITION FUNCTION FOR PERFECT MATCHINGS IN A Hypergraph

by Alexander Barvinok, Alex Samorodnitsky , 2011
"... Given non-negative weights wS on the k-subsets S of a km-element set V, we consider the sum of the products wS1 · · · wSm over all partitions V = S1 ∪... ∪ Sm into pairwise disjoint k-subsets Si. When the weights wS are positive and within a constant factor, fixed in advance, of each other, we pre ..."
Abstract - Cited by 13 (5 self) - Add to MetaCart
Given non-negative weights wS on the k-subsets S of a km-element set V, we consider the sum of the products wS1 · · · wSm over all partitions V = S1 ∪... ∪ Sm into pairwise disjoint k-subsets Si. When the weights wS are positive and within a constant factor, fixed in advance, of each other, we

Rainbow H-factors of complete s-uniform r-partite hypergraphs

by Ailian Chen, Fuji Zhang, Hao Li , 2008
"... We say a s-uniform r-partite hypergraph is complete, if it has a vertex partition {V1, V2,..., Vr} of r classes and its hyperedge set consists of all the s-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete s-uniform r-partite hypergraph with k verti ..."
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We say a s-uniform r-partite hypergraph is complete, if it has a vertex partition {V1, V2,..., Vr} of r classes and its hyperedge set consists of all the s-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete s-uniform r-partite hypergraph with k

How many colors guarantee a rainbow matching?

by Roman Glebov , Benny Sudakov , Tibor Szabó , 2009
"... Abstract Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow t-matching in an r-partite r-uniform multi-hypergraph whose edges are colored with f colors s ..."
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Abstract Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow t-matching in an r-partite r-uniform multi-hypergraph whose edges are colored with f colors
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