Approximation algorithms for the Label-Cover_MAX 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_MAXproblem, and a color-based extension known as the Red-Blue Set Cover problem. First, a randomized algorithm guaranteeing approximation ratio sqrt(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 sqrt(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 sqrt(n log β), where n is the number of sets and β is the number of blue elements. Both algorithms can be adapted to the weighted variants of the respective problems, yielding the same approximation ratios.