Approximating Vertex Cover in Dense Hypergraphs

TL;DR

This paper introduces a randomized polynomial-time algorithm that leverages density and regularity in hypergraphs to surpass the traditional k-approximation barrier for the minimum vertex cover problem.

Contribution

It presents a novel algorithm that improves approximation ratios for dense hypergraphs, extending recursive sampling techniques and establishing optimal bounds under the unique games conjecture.

Findings

Achieves approximation factor k/(1 +(k-1)d/(k Delta))

Provides a best possible approximation of k/(2-1/k) for subdense regular hypergraphs

Breaks the longstanding k-approximation barrier in dense hypergraphs

Abstract

We consider the minimum vertex cover problem in hypergraphs in which every hyperedge has size k (also known as minimum hitting set problem, or minimum set cover with element frequency k). Simple algorithms exist that provide k-approximations, and this is believed to be the best possible approximation achievable in polynomial time. We show how to exploit density and regularity properties of the input hypergraph to break this barrier. In particular, we provide a randomized polynomial-time algorithm with approximation factor k/(1 +(k-1)d/(k Delta)), where d and Delta are the average and maximum degree, respectively, and Delta must be Omega(n^{k-1}/log n). The proposed algorithm generalizes the recursive sampling technique of Imamura and Iwama (SODA'05) for vertex cover in dense graphs. As a corollary, we obtain an approximation factor k/(2-1/k) for subdense regular hypergraphs, which is shown to be the best possible under the unique games conjecture.