Faster algorithms to enumerate hypergraph transversals

TL;DR

This paper introduces new exponential-time algorithms for enumerating all minimal transversals in hypergraphs, improving efficiency for hypergraphs of rank greater than 2 and providing better bounds on their number.

Contribution

The authors develop and analyze faster algorithms for enumerating minimal transversals in hypergraphs of fixed rank, surpassing previous methods in efficiency and bounds.

Findings

Enumerates all minimal transversals in hypergraphs of rank 3 in O(1.6755^n) time.

Improves bounds on the maximum number of minimal transversals for hypergraphs of rank k>2.

Enhances algorithms for counting and finding minimum transversals in hypergraphs of rank k>2 and k>5.

Abstract

A transversal of a hypergraph is a set of vertices intersecting each hyperedge. We design and analyze new exponential-time algorithms to enumerate all inclusion-minimal transversals of a hypergraph. For each fixed k>2, 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>2. 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 on n vertices in time O(1.6755^n). Our algorithm for hypergraphs of rank 4 is based on iterative compression. Our enumeration algorithms improve upon the best known algorithms for counting minimum transversals in hypergraphs of rank k for k>2 and for computing a minimum transversal in hypergraphs of rank k for k>5.