Coverings and matchings in r‐partite hypergraphs

Abstract

AbstractRyser'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