Approximation algorithm for minimum partial multi-cover under a geometric setting

Abstract

In a minimum partial set multi-cover problem (MinPSMC), given an element set X, a collection of subsets ${\mathcal {S}} \subseteq 2^X$, a cost $c_S$ on each set $S\in {\mathcal {S}}$, a covering requirement $r_x$ for each element $x\in X$, and an integer k, the goal is to find a sub-collection ${\mathcal {F}} \subseteq {\mathcal {S}}$ to fully cover at least k elements such that the cost of ${\mathcal {F}}$ is as small as possible, where element x is fully covered by ${\mathcal {F}}$ if it belongs to at least $r_x$ sets of ${\mathcal {F}}$. Recently, it was proved that MinPSMC is at least as hard as the densest k-subgraph problem. The question is: how about the problem in some geometric settings? In this paper, we consider the MinPSMC problem in which X is a set of points on the plane and ${\mathcal {S}}$ is a set of unit squares (MinPSMC-US). Under the assumption that $r_x=f_x$ for every $x\in X$, where $f_x=|\{S\in {\mathcal {S}}:x\in S\}|$ is the number of sets containing element x, we design an approximation algorithm achieving approximation ratio $(1+\varepsilon )$ for MinPSMC-US.