Approximation algorithms for the Label-CoverMAX and Red-Blue Set Cover problems

This paper presents approximation algorithms for two extensions of the set cover problem: a graph-based extension known as the Max-Rep or Label-Cover(MAX)problem, and a color-based extension known as the Red-Blue Set Cover problem. First, a randomized algorithm guaranteeing approximation ratio root n with high probability is proposed for the Max-Rep (or Label-Cover(MAX)) problem, where n is the number of vertices in the graph. This algorithm is then generalized into a 4 root n-ratio algorithm for the nonuniform version of the problem. Secondly, it is shown that the Red-Blue Set Cover problem can be approximated with ratio 2 root n log beta, where n is the number of sets and beta is the number of blue elements. Both algorithms can be adapted to the weighted variants of the respective problems, yielding the same approximation ratios. (C) 2006 Elsevier B.V. All rights reserved.