Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments

TL;DR

This paper introduces a parameterized approximation algorithm for the NP-hard Maximum Weight Independent Set of Rectangles problem, providing new solutions for weighted cases and special segment cases with improved runtime guarantees.

Contribution

It presents the first parameterized approximation algorithms for the weighted MWISR problem, extending techniques from unweighted cases and addressing open problems in the field.

Findings

Provides a $(1- ext{epsilon})$-approximation with runtime $2^{O(k ext{log}(k/ ext{epsilon}))} n^{O(1/ ext{epsilon})}$ for solutions up to size $k$.

Develops a scheme with runtime $2^{O(k^2 ext{log}(k/ ext{epsilon}))} n^{O(1)}$ for segments, achieving near-optimal solutions of size at most $k$.

Addresses open problems in parameterized approximation for weighted MWISR and special segment cases.

Abstract

In the Maximum Weight Independent Set of Rectangles problem (MWISR) we are given a weighted set of $n$ axis-parallel rectangles in the plane. The task is to find a subset of pairwise non-overlapping rectangles with the maximum possible total weight. This problem is NP-hard and the best-known polynomial-time approximation algorithm, due to by Chalermsook and Walczak (SODA 2021), achieves approximation factor $O(\log\log n )$. While in the unweighted setting, constant factor approximation algorithms are known, due to Mitchell (FOCS 2021) and to G\'alvez et al. (SODA 2022), it remains open to extend these techniques to the weighted setting. In this paper, we consider MWISR through the lens of parameterized approximation. Grandoni et al. (ESA 2019) gave a $(1-\epsilon)$-approximation algorithm with running time $k^{O(k/\epsilon^8)} n^{O(1/\epsilon^8)}$ time, where $k$ is the number of rectangles in an optimum solution. Unfortunately, their algorithm works only in the unweighted setting and they left it as an open problem to give a parameterized approximation scheme in the weighted setting. Our contribution is a partial answer to the open question of Grandoni et al. (ESA 2019). We give a parameterized approximation algorithm for MWISR that given a parameter $k$, finds a set of non-overlapping rectangles of weight at least $(1-\epsilon) \text{opt}_k$ in $2^{O(k \log(k/\epsilon))} n^{O(1/\epsilon)}$ time, where $\text{opt}_k$ is the maximum weight of a solution of cardinality at most $k$. Note that thus, our algorithm may return a solution consisting of more than $k$ rectangles. To complement this apparent weakness, we also propose a parameterized approximation scheme with running time $2^{O(k^2 \log(k/\epsilon))} n^{O(1)}$ that finds a solution with cardinality at most $k$ and total weight at least $(1-\epsilon)\text{opt}_k$ for the special case of axis-parallel segments.