Handbook of Combinatorial Designs, Second Edition (Discrete Mathematics and Its Applications)
Handbook of Combinatorial Designs, Second Edition (Discrete Mathematics and Its Applications)
A Comment on Ryser’s Conjecture for Intersecting Hypergraphs
Graphs and Combinatorics
Schema mapping discovery from data instances
Journal of the ACM (JACM)
On linear and semidefinite programming relaxations for hypergraph matching
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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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 1960s, researchers have resolved it for special cases such as for intersecting hypergraphs where r ≤ 5. In this article, we prove several results pertaining to matchings and coverings in r -partite intersecting hypergraphs. First, we prove that finding a minimum cardinality vertex cover for an r -partite intersecting hypergraph is NP-hard. Second, we note Ryser's conjecture for intersecting hypergraphs is easily resolved if a given hypergraph does not contain a particular subhypergraph, which we call a “tornado.” We prove several bounds on the covering number of tornados. Finally, we prove the integrality gap for the standard integer linear programming formulation of the maximum cardinality r -partite hypergraph matching problem is at least r - k where k is the smallest positive integer such that r - k is a prime power. © 2012 Wiley Periodicals, Inc. NETWORKS, Vol. 2012 © 2012 Wiley Periodicals, Inc.