Document Open Access Logo

Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments

Authors Jana Cslovjecsek, Michał Pilipczuk, Karol Węgrzycki

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2024.43.pdf
  • Filesize: 0.77 MB
  • 18 pages

Document Identifiers

Related Versions

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

Cite As Get BibTex

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

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).

References

  1. Anna Adamaszek and Andreas Wiese. Approximation Schemes for Maximum Weight Independent Set of Rectangles. In IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013, pages 400-409, 2013. URL: https://doi.org/10.1109/FOCS.2013.50.
  2. Pankaj K. Agarwal, Marc J. van Kreveld, and Subhash Suri. Label Placement by Maximum Independent Set in Rectangles. Comput. Geom., 11(3-4):209-218, 1998. URL: https://doi.org/10.1016/S0925-7721(98)00028-5.
  3. Brenda S. Baker. Approximation algorithms for NP-complete problems on planar graphs. J. ACM, 41(1):153-180, 1994. URL: https://doi.org/10.1145/174644.174650.
  4. Cristina Bazgan. Schémas d’approximation et complexité paramétrée. Rapport de stage de DEA d’Informatiquea Orsay, 300, 1995. In French. Google Scholar
  5. Paul S. Bonsma, Jens Schulz, and Andreas Wiese. A constant-factor approximation algorithm for unsplittable flow on paths. SIAM J. Comput., 43(2):767-799, 2014. URL: https://doi.org/10.1137/120868360.
  6. Clément Carbonnel, Miguel Romero, and Stanislav Živný. Point-Width and Max-CSPs. ACM Trans. Algorithms, 16(4):54:1-54:28, 2020. URL: https://doi.org/10.1145/3409447.
  7. Marco Cesati and Luca Trevisan. On the efficiency of polynomial time approximation schemes. Information Processing Letters, 64(4):165-171, 1997. URL: https://doi.org/10.1016/S0020-0190(97)00164-6.
  8. Parinya Chalermsook and Bartosz Walczak. Coloring and Maximum Weight Independent Set of Rectangles. In 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 860-868, 2021. URL: https://doi.org/10.1137/1.9781611976465.54.
  9. Julia Chuzhoy and Alina Ene. On Approximating Maximum Independent Set of Rectangles. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, pages 820-829, 2016. URL: https://doi.org/10.1109/FOCS.2016.92.
  10. Jana Cslovjecsek, Michal Pilipczuk, and Karol Wegrzycki. Parameterized approximation for maximum weight independent set of rectangles and segments. CoRR, abs/2212.01620, 2022. URL: https://doi.org/10.48550/arXiv.2212.01620.
  11. Jana Cslovjecsek, Michal Pilipczuk, and Karol Węgrzycki. A polynomial-time OPT^ε-approximation algorithm for maximum independent set of connected subgraphs in a planar graph. In Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, pages 625-638. SIAM, 2024. Google Scholar
  12. Jeffrey S. Doerschler and Herbert Freeman. A rule-based system for dense-map name placement. Commun. ACM, 35(1):68-79, 1992. URL: https://doi.org/10.1145/129617.129620.
  13. Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-Time Approximation Schemes for Geometric Intersection Graphs. SIAM J. Comput., 34(6):1302-1323, 2005. URL: https://doi.org/10.1137/S0097539702402676.
  14. Robert J. Fowler, Michael S. Paterson, and Steven L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12(3):133-137, 1981. URL: https://doi.org/10.1016/0020-0190(81)90111-3.
  15. Eugene C. Freuder. Complexity of k-tree structured constraint satisfaction problems. In 8th National Conference on Artificial Intelligence, pages 4-9, 1990. URL: http://www.aaai.org/Library/AAAI/1990/aaai90-001.php.
  16. Takeshi Fukuda, Yasuhiko Morimoto, Shinichi Morishita, and Takeshi Tokuyama. Data mining with optimized two-dimensional association rules. ACM Trans. Database Syst., 26(2):179-213, 2001. URL: http://portal.acm.org/citation.cfm?id=383891.383893, URL: https://doi.org/10.1145/383891.383893.
  17. Waldo Gálvez, Arindam Khan, Mathieu Mari, Tobias Mömke, Madhusudhan Reddy Pittu, and Andreas Wiese. A 3-Approximation Algorithm for Maximum Independent Set of Rectangles. In 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, pages 894-905, 2022. URL: https://doi.org/10.1137/1.9781611977073.38.
  18. Fabrizio Grandoni, Edin Husić, Mathieu Mari, and Antoine Tinguely. Approximating the Maximum Independent Set of Convex Polygons with a Bounded Number of Directions. Accepted to SoCG 2024, 2024. Google Scholar
  19. Fabrizio Grandoni, Stefan Kratsch, and Andreas Wiese. Parameterized approximation schemes for Independent Set of Rectangles and Geometric Knapsack. In 27th Annual European Symposium on Algorithms, ESA 2019, volume 144 of LIPIcs, pages 53:1-53:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.53.
  20. Jan Kára and Jan Kratochvíl. Fixed Parameter Tractability of Independent Set in Segment Intersection Graphs. In Second International Workshop on Parameterized and Exact Computation, IWPEC 2006, pages 166-174, 2006. URL: https://doi.org/10.1007/11847250_15.
  21. Liane Lewin-Eytan, Joseph Naor, and Ariel Orda. Routing and admission control in networks with advance reservations. In 5th International Workshop on Approximation Algorithms for Combinatorial Optimization, APPROX 2002, pages 215-228, 2002. URL: https://doi.org/10.1007/3-540-45753-4_19.
  22. Dániel Marx. Efficient approximation schemes for geometric problems? In 13th Annual European Symposium on Algorithms, ESA 2005, pages 448-459, 2005. URL: https://doi.org/10.1007/11561071_41.
  23. Dániel Marx. Parameterized Complexity of Independence and Domination on Geometric Graphs. In Second International Workshop on Parameterized and Exact Computation, IWPEC 2006, pages 154-165, 2006. URL: https://doi.org/10.1007/11847250_14.
  24. Joseph S. B. Mitchell. Approximating Maximum Independent Set for rectangles in the plane. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, pages 339-350, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00042.
  25. Neil Robertson and Paul D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  26. Miguel Romero, Marcin Wrochna, and Stanislav Živný. Treewidth-Pliability and PTAS for Max-CSPs. In 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 473-483, 2021. URL: https://doi.org/10.1137/1.9781611976465.29.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact