On Approximating Maximum Independent Set of Rectangles

TL;DR

This paper advances the approximation algorithms for the Maximum Independent Set of Rectangles problem by providing a near-polynomial time algorithm that achieves a (1 - ε)-approximation, improving upon previous results and introducing new techniques.

Contribution

It presents a new algorithm that achieves a (1 - ε)-approximation for MISR in quasi-polynomial time, moving closer to a polynomial-time PTAS.

Findings

Achieves (1 - ε)-approximation in n^{O(poly(log log n / ε))} time

Introduces new technical ideas for geometric approximation problems

Progress towards polynomial-time approximation schemes for MISR

Abstract

We study the Maximum Independent Set of Rectangles (MISR) problem: given a set of $n$ axis-parallel rectangles, find a largest-cardinality subset of the rectangles, such that no two of them overlap. MISR is a basic geometric optimization problem with many applications, that has been studied extensively. Until recently, the best approximation algorithm for it achieved an $O(\log \log n)$-approximation factor. In a recent breakthrough, Adamaszek and Wiese provided a quasi-polynomial time approximation scheme: a $(1-\epsilon)$-approximation algorithm with running time $n^{O(\operatorname{poly}(\log n)/\epsilon)}$. Despite this result, obtaining a PTAS or even a polynomial-time constant-factor approximation remains a challenging open problem. In this paper we make progress towards this goal by providing an algorithm for MISR that achieves a $(1 - \epsilon)$-approximation in time $n^{O(\operatorname{poly}(\log\log{n} / \epsilon))}$. We introduce several new technical ideas, that we hope will lead to further progress on this and related problems.