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 exponential-time polynomial-space algorithms to enumerate all inclusion-minimal transversals of a hypergraph. For each fixed $$k\ge 3$$ , 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 $$k\ge 3$$ . 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 $$O(1.6755^n)$$ . Our enumeration algorithms improve upon the best known algorithms for counting minimum transversals in hypergraphs of rank k for $$k\ge 3$$ and for computing a minimum transversal in hypergraphs of rank k for $$k\ge 6$$ .