Approximation Schemes for Geometric Coverage Problems

TL;DR

This paper develops a PTAS for geometric maximum coverage problems with cardinality constraints by extending planar separator techniques, confirming conjectures for 3D halfspaces, and broadening the applicability of local search methods.

Contribution

It introduces a color-balanced planar separator theorem and applies it to obtain a PTAS for maximum coverage with cardinality constraints in planarizable instances.

Findings

Provides a PTAS for maximum k-coverage with cardinality constraints in planarizable instances.

Confirms the conjecture for three-dimensional real halfspaces.

Extends local search techniques to more complex geometric coverage problems.

Abstract

In their seminal work, Mustafa and Ray (2009) showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search -- this is one of the most general approaches known for such problems. Their result applies if a naturally defined "exchange graph" for two feasible solutions is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson (1987). Obtaining similar results for the related maximum k-coverage problem (MC) seems non-trivial due to the hard cardinality constraint. In fact, while Badanidiyuru, Kleinberg, and Lee (2012) have shown (via a different analysis) that local search yields a PTAS for two-dimensional real halfspaces, they only conjectured that the same holds true for dimension three. Interestingly, at this point it was already known that local search provides a PTAS for the corresponding set cover case and this followed directly from the approach of Mustafa and Ray. In this work we provide a way to address the above-mentioned issue. First, we propose a color-balanced version of the planar separator theorem. The resulting subdivision approximates locally in each part the global distribution of the colors. Second, we show how this roughly balanced subdivision can be employed in a more careful analysis to strictly obey the hard cardinality constraint. More specifically, we obtain a PTAS for any "planarizable" instance of MC and thus essentially for all cases where the corresponding SC instance can be tackled via the approach of Mustafa and Ray. As a corollary, we confirm the conjecture of Badanidiyuru, Kleinberg, and Lee regarding real half spaces in dimension three. We feel that our ideas could also be helpful in other geometric settings involving a cardinality constraint.