eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-23
43:1
43:18
10.4230/LIPIcs.ESA.2024.43
article
Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments
Cslovjecsek, Jana
1
Pilipczuk, Michał
2
Węgrzycki, Karol
3
EPFL, Lausanne, Switzerland
Institute of Informatics, University of Warsaw, Poland
Saarland University and Max Planck Institute for Informatics, Saarbrücken, Germany
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 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 Gálvez 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, Kratsch and Wiese [ESA 2019] gave a (1-ε)-approximation algorithm running in k^{𝒪(k/ε⁸)} n^{𝒪(1/ε⁸)} 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.
We give a parameterized approximation algorithm for MWISR that given a parameter k ∈ ℕ, finds a set of non-overlapping rectangles of weight at least (1-ε) opt_k in 2^{𝒪(k log(k/ε))} n^{𝒪(1/ε)} time, where opt_k is the maximum weight of a solution of cardinality at most k. We also propose a parameterized approximation scheme with running time 2^{𝒪(k² log(k/ε))} n^{𝒪(1)} that finds a solution with cardinality at most k and total weight at least (1-ε)opt_k for the special case of axis-parallel segments.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol308-esa2024/LIPIcs.ESA.2024.43/LIPIcs.ESA.2024.43.pdf
parameterized approximation
Maximum Weight Independent Set
rectangles
segments