On Partial Covering For Geometric Set Systems

TL;DR

This paper introduces an LP rounding approach for the Partial Set Cover problem in geometric set systems, providing improved approximation guarantees by relating the integrality gap to that of the classic Set Cover LP.

Contribution

It presents a novel LP rounding scheme that bounds the integrality gap of Partial Set Cover in geometric set systems, enhancing approximation results.

Findings

LP rounding scheme for Partial Set Cover

Bound on integrality gap proportional to Set Cover LP

Improved approximation guarantees for geometric set systems

Abstract

We study a generalization of the Set Cover problem called the \emph{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.