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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∆ ), whered and ∆ are the average and maximum degree, respectively, and ∆ must be Ω(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.

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We study approximability of subdense instances of various covering problems on graphs, defined as instances in which the minimum or average degree is Ω(n/ψ(n)) for some function ψ(n) = ω(1) of the instance size. We design new approximation algorithms as well as new polynomial time approximation schemes (PTASs) for those problems and establish first approximation hardness results for them. Interestingly, in some cases we were able to prove optimality of the underlying approximation ratios, under usual complexity-theoretic assumptions. Our results for the Vertex Cover problem depend on an improved recursive sampling method which could be of independent interest.