  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 $P_1,..., P_r$ of the edge set $E$, and a parameter $k_i$
for each partition $P_i$. The goal is to find a minimum cost set of vertices
which cover at least $k_i$ edges from the partition $P_i$. 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.
