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 <= \le |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^d, 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.
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, \mathcal{S ), where X is a set of elements and \mathcal{S is a collection of subsets of X, and an integer k \le |X|. The goal is to cover at least k elements of X by using a minimum-weight collection of sets from \mathcal{S 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 a certain projection of the set system (X, \mathcal{S . 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.
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S. In the geometric version, X typically consists of points in R^d, and S contains sets induced by geometric objects of a certain kind.
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a certain projection of the set system (X, \mathcal{S