On Partial Covering For Geometric Set Systems

We study a generalization of the Set Cover problem called the Partial Set Cover in the context of geometric set systems. The input to this problem is a set system (X ,S), where X is a set of elements and S is a collection of subsets of X , and an integer k ≤ |X |. The goal is to cover at least k elements of X by using a minimum-weight collection of sets from S. In the geometric version, X typically consists of points in R, and S contains sets induced by geometric objects of a certain kind. The main result of this article is an LP rounding scheme which shows that the integrality gap of the Partial Set Cover LP is at most a constant times that of the Set Cover LP for the same class of geometric objects. As a corollary of this result, we get improved approximation guarantees for the Partial Set Cover problem for a large class of geometric set systems.