Faster algorithms to enumerate hypergraph transversals

Abstract

A transversal of a hypergraph is a set of vertices intersecting each hyperedge. We design and analyze new exponentialtime polynomial-space algorithms to enumerate all inclusion-minimal transversals of a hypergraph. For each fixed k ≥ 3, our algorithms for hypergraphs of rank k, where the rank is the maximum size of a hyperedge, outperform the previous best. This also implies improved upper bounds on the maximum number of minimal transversals in n-vertex hypergraphs of rank k ≥ 3. Our main algorithm is a branching algorithm whose running time is analyzed with Measure and Conquer. It enumerates all minimal transversals of hypergraphs of rank 3 in time O(1.6755n). Our enumeration algorithms improve upon the best known algorithms for counting minimum transversals in hypergraphs of rank k for k ≥ 3 and for computing a minimum transversal in hypergraphs of rank k for k ≥ 6.