Minimum Membership Geometric Set Cover in the Continuous Setting

2025-05-30

AI Summary

This paper studies the Minimum Membership Geometric Set Cover (MMGSC) problem in continuous space: given a set of points in ℝ² and a family of geometric objects, find a minimum-membership cover—i.e., a collection of translated copies covering all points while minimizing the maximum number of objects covering any single point. We present the first optimal 1-membership cover algorithm for unit squares, running in O(n log n) time and yielding solutions at most twice the optimal size; characterize the class of geometric objects admitting 1-membership covers; prove that unit disks always admit 2-membership covers and provide a constructive algorithm achieving this with disks scaled by factor 7; extend results to d-dimensional hypercubes, attaining a 2^{d−1}-approximation; and design an O(n log n + nm)-time 4-membership cover algorithm for convex m-gons. The core contribution is the first systematic theoretical framework and efficient algorithms for membership-constrained geometric covering under continuous translations.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $mathsf{memb}(P, mathcal{S})=max_{pin P}|{sin mathcal{S}: pin s}|$. For unit squares, we present a simple $O(nlog n)$ 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 $O(dnlog n)$ 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 $O(nlog n + nm)$ time.

Problem

• Minimize overlapping geometric object coverage for point sets
• Develop efficient algorithms for unit squares and hyperboxes
• Analyze membership cover bounds for disks and convex polygons

Innovation

• O(n log n) algorithm for 1-membership cover
• NP-hardness proof for non-overlapping squares
• Generalization to d-dimensional hyperboxes