Approximation Schemes for Covering and Packing

TL;DR

This paper extends the local search framework for PTASs to broader graph families and introduces new applications, including geometric covering and packing problems, with improved solutions for specific cases.

Contribution

It generalizes the existing framework to more graph classes and applies it to novel geometric problems, achieving PTASs for complex covering and packing tasks.

Findings

PTASs for maximum l-shallow set of fat objects

PTAS for maximum triangle matching in l-shallow unit disk graphs

Improved PTAS for minimum disk cover of points

Abstract

The local search framework for obtaining PTASs for NP-hard geometric optimization problems was introduced, independently, by Chan and Har-Peled (2009) and Mustafa and Ray (2010). In this paper, we generalize the framework by extending its analysis to additional families of graphs, beyond the family of planar graphs. We then present several applications of the generalized framework, some of which are very different from those presented to date (using the original framework). These applications include PTASs for finding a maximum l-shallow set of a set of fat objects, for finding a maximum triangle matching in an l-shallow unit disk graph, and for vertex-guarding a (not-necessarily-simple) polygon under an appropriate shallowness assumption. We also present a PTAS (using the original framework) for the important problem where one has to find a minimum-cardinality subset of a given set of disks (of varying radii) that covers a given set of points, and apply it to a class cover problem (studied in Bereg et al., 2012) to obtain an improved solution.