Approximation Algorithms for Maximum Independent Set of Pseudo-Disks

TL;DR

This paper develops approximation algorithms for the maximum independent set problem in pseudo-disks, providing a PTAS for unweighted cases and a constant-factor approximation for weighted cases, using novel combinatorial techniques.

Contribution

It introduces new approximation algorithms for pseudo-disk independent sets, including a PTAS for unweighted and a novel LP-based scheme for weighted cases.

Findings

PTAS achieved for unweighted maximum independent set of pseudo-disks

Constant-factor approximation for weighted case using LP relaxation

New combinatorial analysis techniques developed

Abstract

We present approximation algorithms for maximum independent set of pseudo-disks in the plane, both in the weighted and unweighted cases. For the unweighted case, we prove that a local search algorithm yields a \PTAS. For the weighted case, we suggest a novel rounding scheme based on an \LP relaxation of the problem, which leads to a constant-factor approximation. Most previous algorithms for maximum independent set (in geometric settings) relied on packing arguments that are not applicable in this case. As such, the analysis of both algorithms requires some new combinatorial ideas, which we believe to be of independent interest.