Approximation algorithms for the partition vertex cover problem

We consider a natural generalization of the Partial Vertex Cover problem. Here an instance consists of a graph G=(V,E)G=(V,E), a cost function c:V→Z+c:V→Z+, a partition P1,…,PrP1,…,Pr of the edge set E  , and a parameter kiki for each partition PiPi. The objective is to find a minimum cost set of vertices which cover at least kiki edges from the partition PiPi. 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)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. For example we consider a problem where additionally edges have profits, and we would like to pick a minimum cost set of vertices which cover edges of total profit at least ΠiΠi for each partition PiPi. We call this the Knapsack Partition Vertex Cover problem. We give an O(log⁡r)O(log⁡r) approximation algorithm for this problem as well.