Approximation algorithms for minimum ply covering of points with unit squares and unit disks

Abstract

Given a set P of points and a set U of geometric objects in the Euclidean plane, a minimum ply cover of P with U is a subset of U that covers P and minimizes the number of objects that share a common intersection, called the minimum ply cover number of P with U. Biedl et al. (2021) [9] showed that for both unit squares and unit disks, determining the minimum ply cover number for a set of points is NP-hard. They gave polynomial-time 2-approximation algorithms for the special case when the minimum ply cover number is constant, and asked whether there exists polynomial-time $O\left(1\right)$-approximation algorithms for these problems. In this paper, we settle the question posed by Biedl et al. by providing polynomial-time $O\left(1\right)$-approximation algorithms for the minimum ply cover problem for both unit squares and unit disks.