Approximating vertex cover in dense hypergraphs

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 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¯kΔ), where d¯ and Δ are the average and maximum degree, and Δ must be Ω(nk−1/logn). The proposed algorithm generalizes the recursive sampling technique of Imamura and Iwama (SODAʼ05) for vertex cover in dense graphs. As a corollary, we obtain an approximation factor arbitrarily close to k/(2−1/k)k/(2−1/k) for subdense regular hypergraphs, which is shown to be the best possible under the Unique Games conjecture.