Approximation Algorithm for Minimum $p$ Union Under a Geometric Setting

TL;DR

This paper introduces a bicriteria approximation algorithm for a geometric version of the minimum p union problem, where the goal is to cover as few points as possible with a limited number of unit squares, providing near-optimal solutions.

Contribution

It presents the first approximation algorithm for the geometric Min p Union problem, achieving a bicriteria approximation with guarantees on the number of squares and points covered.

Findings

Provides a (rac{1}{1+ ext{epsilon}},4)-bicriteria approximation algorithm.

Achieves near-optimal coverage with theoretical guarantees.

Extends the understanding of geometric set cover problems.

Abstract

In a minimum $p$ union problem (Min$p$U), given a hypergraph $G=(V,E)$ and an integer $p$, the goal is to find a set of $p$ hyperedges $E'\subseteq E$ such that the number of vertices covered by $E'$ (that is $|\bigcup_{e\in E'}e|$) is minimized. It was known that Min$p$U is at least as hard as the densest $k$-subgraph problem. A question is: how about the problem in some geometric settings? In this paper, we consider the unit square Min$p$U problem (Min$p$U-US) in which $V$ is a set of points on the plane, and each hyperedge of $E$ consists of a set of points in a unit square. A $(\frac{1}{1+\varepsilon},4)$-bicriteria approximation algorithm is presented, that is, the algorithm finds at least $\frac{p}{1+\varepsilon}$ unit squares covering at most $4opt$ points, where $opt$ is the optimal value for the Min$p$U-US instance (the minimum number of points that can be covered by $p$ unit squares).