On approximation of the vertex cover problem in hypergraphs

This paper deals with approximation of the vertex cover problem in hypergraphs with bounded degree and bounded number of neighboring vertices. For hypergraphs with edges of size at most r and degree bounded by Δ we extend a result of Krivelevich and obtain a ⌈βr⌉ approximation algorithm, where 0<β<1 satisfies 1-β=[βr/(βr+1)]Δ-1/βr. In particular, we show that when (logΔ)/r⩾1-1/e the approximation guarantee of our algorithm is better than that of the greedy algorithm. For hypergraphs in which each vertex has at most D adjacent vertices and its degree is bounded by Δ⩾D, we show that the greedy heuristic provides an H(Δ,D)⩽(D-1)[1-Δ1/(1-D)]+1 approximation, which in some cases significantly improves the well known H(Δ)⩽logΔ+1 bound.