Minimum Membership Geometric Set Cover in the Continuous Setting

TL;DR

This paper investigates the minimum membership geometric set cover problem in the continuous setting, providing efficient algorithms and approximation bounds for various geometric objects such as squares, disks, and convex polygons.

Contribution

It introduces a simple $O(n ext{log}n)$ algorithm for unit squares with a 1-membership cover, proves NP-hardness for non-overlapping squares, and extends results to hyperboxes and other convex objects.

Findings

A $O(n ext{log}n)$ algorithm for 1-membership cover with at most twice the optimal size for unit squares.

NP-hardness of covering points with non-overlapping unit squares.

Existence of bounded 1-membership covers for certain classes of objects.

Abstract

We study the minimum membership geometric set cover, i.e., MMGSC problem [SoCG, 2023] in the continuous setting. In this problem, the input consists of a set $P$ of $n$ points in $\mathbb{R}^{2}$, and a geometric object $t$, the goal is to find a set $\mathcal{S}$ of translated copies of the geometric object $t$ that covers all the points in $P$ while minimizing $\mathsf{memb}(P, \mathcal{S})$, where $\mathsf{memb}(P, \mathcal{S})=\max_{p\in P}|\{s\in \mathcal{S}: p\in s\}|$. For unit squares, we present a simple $O(n\log n)$ time algorithm that outputs a $1$-membership cover. We show that the size of our solution is at most twice that of an optimal solution. We establish the NP-hardness on the problem of computing the minimum number of non-overlapping unit squares required to cover a given set of points. This algorithm also generalizes to fixed-sized hyperboxes in $d$-dimensional space, where an $1$-membership cover with size at most $2^{d-1}$ times the size of a minimum-sized $1$-membership cover is computed in $O(dn\log n)$ time. Additionally, we characterize a class of objects for which a $1$-membership cover always exists. For unit disks, we prove that a $2$-membership cover exists for any point set, and the size of the cover is at most $7$ times that of the optimal cover. For arbitrary convex polygons with $m$ vertices, we present an algorithm that outputs a $4$-membership cover in $O(n\log n + nm)$ time.