Coverings and matchings in r-partite hypergraphs

Ryser's conjecture postulates that for r -partite hypergraphs, t = (r - 1)? where t is the covering number of the hypergraph and ? 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. (c) 2012 Wiley Periodicals, Inc. NETWORKS, Vol. 2012