A QPTAS for Maximum Weight Independent Set of Polygons with Polylogarithmically Many Vertices

TL;DR

This paper introduces a quasi-polynomial time approximation scheme for the Maximum Weight Independent Set of Polygons problem, achieving near-optimal solutions when polygons have polylogarithmic vertices, extending existing geometric approximation frameworks.

Contribution

It extends a geometric approximation framework to handle polygons with arbitrary angles and polylogarithmic vertices, providing a QPTAS for the problem.

Findings

Achieves a (1+epsilon)-approximation with quasi-polynomial time.

Reduces the core problem to triangles for technical simplicity.

Develops a new plane partitioning method handling arbitrary angles.

Abstract

The Maximum Weight Independent Set of Polygons problem is a fundamental problem in computational geometry. Given a set of weighted polygons in the 2-dimensional plane, the goal is to find a set of pairwise non-overlapping polygons with maximum total weight. Due to its wide range of applications, the MWISP problem and its special cases have been extensively studied both in the approximation algorithms and the computational geometry community. Despite a lot of research, its general case is not well-understood. Currently the best known polynomial time algorithm achieves an approximation ratio of n^(epsilon) [Fox and Pach, SODA 2011], and it is not even clear whether the problem is APX-hard. We present a (1+epsilon)-approximation algorithm, assuming that each polygon in the input has at most a polylogarithmic number of vertices. Our algorithm has quasi-polynomial running time. We use a recently introduced framework for approximating maximum weight independent set in geometric intersection graphs. The framework has been used to construct a QPTAS in the much simpler case of axis-parallel rectangles. We extend it in two ways, to adapt it to our much more general setting. First, we show that its technical core can be reduced to the case when all input polygons are triangles. Secondly, we replace its key technical ingredient which is a method to partition the plane using only few edges such that the objects stemming from the optimal solution are evenly distributed among the resulting faces and each object is intersected only a few times. Our new procedure for this task is not more complex than the original one, and it can handle the arising difficulties due to the arbitrary angles of the polygons. Note that already this obstacle makes the known analysis for the above framework fail. Also, in general it is not well understood how to handle this difficulty by efficient approximation algorithms.