Approximation Algorithms for Edge Partitioned Vertex Cover Problems

We consider a natural generalization of the Partial Vertex Cover problem. Here an instance consists of a graph G = (V,E), a cost function c : V → Z, a partition P1, . . . , Pr of the edge set E, and a parameter ki for each partition Pi. The goal is to find a minimum cost set of vertices which cover at least ki edges from the partition Pi. We call this the Partition-VC problem. In this paper, we give matching upper and lower bound on the approximability of this problem. Our algorithm is based on a novel LP relaxation for this problem. This LP relaxation is obtained by adding knapsack cover inequalities to a natural LP relaxation of the problem. We show that this LP has integrality gap of O(log r), where r is the number of sets in the partition of the edge set. We also extend our result to more general settings.