Minimum Ply Covering of Points with Unit Squares

TL;DR

This paper presents a polynomial-time approximation algorithm with a factor of (8+ε) for the minimum ply covering problem of points with unit squares, resolving an open question about its approximability.

Contribution

It introduces the first polynomial-time approximation algorithm with a constant factor for the general case of the minimum ply cover problem.

Findings

Provides a polynomial-time (8+ε)-approximation algorithm for the problem.

Resolves the open question of approximability when the minimum ply cover number is unbounded.

Extends previous results limited to constant ply cover numbers.

Abstract

Given a set $P$ of points and a set $U$ of axis-parallel unit squares 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 squares that share a common intersection, called the minimum ply cover number of $P$ with $U$. Biedl et al. [Comput. Geom., 94:101712, 2020] showed that determining the minimum ply cover number for a set of points by a set of axis-parallel unit squares is NP-hard, and gave a polynomial-time 2-approximation algorithm for instances in which the minimum ply cover number is constant. The question of whether there exists a polynomial-time approximation algorithm remained open when the minimum ply cover number is $\omega(1)$. We settle this open question and present a polynomial-time $(8+\varepsilon)$-approximation algorithm for the general problem, for every fixed $\varepsilon>0$.