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 positive 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 Vertex Cover 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.