Coverings and matchings in r-partite hypergraphs

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 paper, 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 sub-hypergraph, 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.