A sharp bound on the integrality gap in the 3-set cover problem

Abstract

Given a hypergraph with edges of size at most 3, the 3-set cover problem asks to determine the minimum size of a family of edges which covers the vertex set. As the problem is NP-hard, it is natural to consider its fractional (linear programming) relaxation, which provides a lower bound on the value of the optimal solution. The ratio between the actual value and that of the fractional relaxation is called the integrality gap. A classic bound of Lovász implies that the integrality gap in this problem is at most 11/6. This has been improved to 5/3 by Fujito and Okumura. Here we prove that the integrality gap is at most 3/2, which is best possible. A corollary of this result is that the vertex set of any 3-uniform, regular hypergraph on n vertices can be covered by n/2 (or fewer) edges. This solves the k=3 case of a problem of de A. Moreira and Kohayakawa. As another application, we derive a certain variant of the Gale-Shapley stable marriage theorem for triples.