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Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments

Authors Jana Cslovjecsek, Michał Pilipczuk, Karol Węgrzycki

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Author Details

Jana Cslovjecsek
  • EPFL, Lausanne, Switzerland
Michał Pilipczuk
  • Institute of Informatics, University of Warsaw, Poland
Karol Węgrzycki
  • Saarland University and Max Planck Institute for Informatics, Saarbrücken, Germany

Funding

This work is a part of projects BOBR (Michał Pilipczuk) and TIPEA (Karol Węgrzycki) that have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements no. 948057 and 850979, respectively).

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Jana Cslovjecsek, Michał Pilipczuk, and Karol Węgrzycki. Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.43

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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • parameterized approximation
  • Maximum Weight Independent Set
  • rectangles
  • segments

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References

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