Quasi-Polynomial Time Approximation Scheme for Sparse Subsets of Polygons

TL;DR

This paper presents a quasi-polynomial time approximation scheme for finding large sparse subsets of polygons, extending previous work to polygons of arbitrary complexity and specific sparsity conditions like cycle-free intersection graphs.

Contribution

It introduces a quasi-polynomial time algorithm for approximating the largest independent set of polygons with arbitrary complexity and specific sparsity constraints.

Findings

Algorithm works for polygons of arbitrary complexity.

Approximates largest subset with no 4-cycle in intersection graph.

Extends previous results to broader classes of polygons.

Abstract

We describe how to approximate, in quasi-polynomial time, the largest independent set of polygons, in a given set of polygons. Our algorithm works by extending the result of Adamaszek and Wiese \cite{aw-asmwi-13, aw-qmwis-14} to polygons of arbitrary complexity. Surprisingly, the algorithm also works or computing the largest subset of the given set of polygons that has some sparsity condition. For example, we show that one can approximate the largest subset of polygons, such that the intersection graph of the subset does not contain a cycle of length $4$ (i.e., $K_{2,2}$).