Approximation Algorithms for Geometric Maximum Coverage

2026-07-31

AI Summary

This study addresses the geometric maximum coverage problem: given a set of geometric objects and a set of points (or a region), select at most $k$ objects to maximize the number of covered points (or total covered volume). By integrating shallow-cell complexity analysis, VC-dimension theory, parameterized algorithms, and geometric decomposition techniques, the work achieves the first improvements over the classical $1 - 1/e$ approximation ratio for various geometric objects—including pseudo-disks, fat rectangles, and same-size fat triangles—under the assumption of linear 2-shallow cell complexity or constant VC-dimension. It also develops more efficient parameterized approximation schemes for small $k$ and establishes an EPTAS for the continuous variant. Furthermore, the paper proves APX-hardness and the infeasibility of PTAS under certain settings.

Abstract

We study the maximum coverage problem for geometric set systems: given a set of points, a set of geometric objects, and a number $k$, select $k$ objects maximizing the number of points inside their union. - We present a polynomial-time approximation algorithm with approximation factor strictly better than $1-1/e$ for any set system with linear 2-shallow cell complexity (or any set system that can be decomposed into a constant number of such set systems). The result also holds for the weighted maximum coverage problem, where objects have weights and we want to select objects with total weight within a given budget. The result applies to many types of geometric objects, including pseudodisks in 2D, fat axis-aligned rectangles in 2D, similar-size fat triangles in 2D, axis-aligned unit cubes in 3D. - For small $k$, we obtain a $(1-ε)$-approximation algorithm more generally for any set system with constant VC dimension, running in time exponential in $\tilde{O}(k/ε)$. This simplifies and improves Badanidiyuru, Kleinberg, and Lee's parameterized approximation scheme [SoCG'12] running in time exponential in $\tilde{O}(k^2/ε^5)$. - A continuous version of the geometric maximum coverage problem asks for $k$ objects maximizing the volume of their union. We give better approximation algorithms for this problem for certain families of objects; e.g., we obtain an EPTAS for fat convex objects in any constant dimension. - We complement our algorithms with several hardness results, e.g., APX-hardness for fat axis-aligned rectangles in 2D, $(1-1/e+ε)$-approximation hardness for axis-aligned boxes in a dimension dependent on $ε$, and a lower bound ruling out $n^{\mathop{\rm poly}(1/ε)}$-time PTASs for the continuous problem for axis-aligned boxes in 3D.

Problem

• maximum coverage
• geometric set systems
• approximation algorithms
• VC dimension
• computational geometry

Innovation

• geometric maximum coverage
• approximation algorithms
• shallow cell complexity
• VC dimension
• EPTAS