Approximate Set Covering in Uniform Hypergraphs

The weighted set covering problem, restricted to the class of r-uniform hypergraphs, is considered. We propose a new approach, based on a recent result of Aharoni, Holzman, and Krivelevich about the ratio of integer and fractional covering numbers in k-colorable r-uniform hypergraphs. This approach, applied to hypergraphs of maximal degree bounded by Δ, yields an algorithm with approximation ratio r(1 - c/Δ^1/(r-1)). Next, we combine this approach with an adaptation of the local ratio theorem of Bar-Yehuda and Even for hypergraphs and present a general framework of approximation algorithms, based on subhypergraph exclusion. An application of this scheme is described, providing an algorithm with approximation ratio r(1 - c/n^(r-1)/r) for hypergraphs on n vertices. We discuss also the limitations of this approach.