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Approximating the Maximum Independent Set of Convex Polygons with a Bounded Number of Directions

Authors Fabrizio Grandoni , Edin Husić , Mathieu Mari , Antoine Tinguely

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Author Details

Fabrizio Grandoni
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Edin Husić
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Mathieu Mari
  • LIRMM, University of Montpellier, CNRS, Montpellier, France
Antoine Tinguely
  • IDSIA, USI-SUPSI, Lugano, Switzerland

Funding

  • Grandoni, Fabrizio: Partially supported by the Swiss National Science Foundation (SNSF) Grant 200021 200731/1.
  • Husić, Edin: Supported by the Swiss National Science Foundation (SNSF) Grant 200021 200731/1.
  • Tinguely, Antoine: Supported by the Swiss National Science Foundation (SNSF) Grant 200021 200731/1.

Cite AsGet BibTex

Fabrizio Grandoni, Edin Husić, Mathieu Mari, and Antoine Tinguely. Approximating the Maximum Independent Set of Convex Polygons with a Bounded Number of Directions. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.61

Abstract

In the maximum independent set of convex polygons problem, we are given a set of n convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the problem where the edges of the polygons can take at most d fixed directions. We present an 8d/3-approximation algorithm for this problem running in time O((nd)^O(d4^d)). The previous-best polynomial-time approximation (for constant d) was a classical n^ε approximation by Fox and Pach [SODA'11] that has recently been improved to a OPT^ε-approximation algorithm by Cslovjecsek, Pilipczuk and Węgrzycki [SODA '24], which also extends to an arbitrary set of convex polygons. Our result builds on, and generalizes the recent constant factor approximation algorithms for the maximum independent set of axis-parallel rectangles problem (which is a special case of our problem with d = 2) by Mitchell [FOCS'21] and Gálvez, Khan, Mari, Mömke, Reddy, and Wiese [SODA'22].

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Theory of computation → Computational geometry
Keywords
  • Approximation algorithms
  • packing
  • independent set
  • polygons

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References

  1. Anna Adamaszek, Sariel Har-Peled, and Andreas Wiese. Approximation schemes for independent set and sparse subsets of polygons. J. ACM, 66(4):29:1-29:40, 2019. URL: https://doi.org/10.1145/3326122.
  2. Anna Adamaszek and Andreas Wiese. A quasi-ptas for the two-dimensional geometric knapsack problem. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1491-1505. SIAM, 2015. URL: https://doi.org/10.1137/1.9781611973730.98.
  3. Aris Anagnostopoulos, Fabrizio Grandoni, Stefano Leonardi, and Andreas Wiese. A mazing 2+ε approximation for unsplittable flow on a path. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 26-41. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.3.
  4. Nikhil Bansal, Amit Chakrabarti, Amir Epstein, and Baruch Schieber. A quasi-ptas for unsplittable flow on line graphs. In Jon M. Kleinberg, editor, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 721-729. ACM, 2006. URL: https://doi.org/10.1145/1132516.1132617.
  5. Nikhil Bansal, Zachary Friggstad, Rohit Khandekar, and Mohammad R. Salavatipour. A logarithmic approximation for unsplittable flow on line graphs. ACM Trans. Algorithms, 10(1):1:1-1:15, 2014. URL: https://doi.org/10.1145/2532645.
  6. 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.
  7. Parinya Chalermsook and Julia Chuzhoy. Maximum independent set of rectangles. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 892-901. SIAM, 2009. URL: http://dl.acm.org/citation.cfm?id=1496770.1496867.
  8. Parinya Chalermsook and Bartosz Walczak. Coloring and maximum weight independent set of rectangles. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 860-868. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.54.
  9. Timothy M Chan. Polynomial-time approximation schemes for packing and piercing fat objects. Journal of Algorithms, 46(2):178-189, 2003. Google Scholar
  10. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discret. Comput. Geom., 48(2):373-392, 2012. URL: https://doi.org/10.1007/s00454-012-9417-5.
  11. Hsien-Chih Chang, Jeff Erickson, and Chao Xu. Detecting weakly simple polygons. In Proceedings of the twenty-sixth annual ACM-SIAM Symposium on Discrete Algorithms, pages 1655-1670. SIAM, 2014. Google Scholar
  12. Jana Cslovjecsek, Michał 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 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 625-638. SIAM, 2024. Google Scholar
  13. Leila De Floriani, Paola Magillo, and Enrico Puppo. Applications of computational geometry to geographic information systems. Handbook of computational geometry, 7:333-388, 2000. Google Scholar
  14. Erik D Demaine and Joseph O'Rourke. Geometric folding algorithms: linkages, origami, polyhedra. Cambridge university press, 2007. Google Scholar
  15. 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.
  16. 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. Google Scholar
  17. Jacob Fox and János Pach. Computing the independence number of intersection graphs. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1161-1165. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.87.
  18. Takeshi Fukuda, Yasuhiko Morimoto, Shinichi Morishita, and Takeshi Tokuyama. Data mining with optimized two-dimensional association rules. ACM Transactions on Database Systems (TODS), 26(2):179-213, 2001. Google Scholar
  19. Waldo Gálvez, Fabrizio Grandoni, Salvatore Ingala, Sandy Heydrich, Arindam Khan, and Andreas Wiese. Approximating geometric knapsack via l-packings. ACM Trans. Algorithms, 17(4):33:1-33:67, 2021. URL: https://doi.org/10.1145/3473713.
  20. Waldo Gálvez, Fabrizio Grandoni, Arindam Khan, Diego Ramírez-Romero, and Andreas Wiese. Improved approximation algorithms for 2-dimensional knapsack: Packing into multiple l-shapes, spirals, and more. In 37th International Symposium on Computational Geometry (SoCG), volume 189, pages 39:1-39:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.SoCG.2021.39.
  21. 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 Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 894-905. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.38.
  22. Waldo Gálvez, Arindam Khan, Mathieu Mari, Tobias Mömke, Madhusudhan Reddy, and Andreas Wiese. A (2+ε)-approximation algorithm for maximum independent set of rectangles. arXiv preprint, 2021. URL: https://arxiv.org/abs/2106.00623.
  23. Fabrizio Grandoni, Edin Husić, Mathieu Mari, and Antoine Tinguely. Approximating the maximum independent set of convex polygons with a bounded number of directions. arXiv preprint, 2024. URL: https://arxiv.org/abs/2402.07666.
  24. Fabrizio Grandoni, Stefan Kratsch, and Andreas Wiese. Parameterized approximation schemes for independent set of rectangles and geometric knapsack. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, 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.
  25. Fabrizio Grandoni, Tobias Mömke, and Andreas Wiese. A PTAS for unsplittable flow on a path. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 289-302. ACM, 2022. URL: https://doi.org/10.1145/3519935.3519959.
  26. Fabrizio Grandoni, Tobias Mömke, and Andreas Wiese. Unsplittable flow on a path: The game! In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 906-926. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.39.
  27. Fabrizio Grandoni, Tobias Mömke, Andreas Wiese, and Hang Zhou. A (5/3 + ε)-approximation for unsplittable flow on a path: placing small tasks into boxes. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 607-619. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188894.
  28. Dorit S Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and vlsi. Journal of the ACM (JACM), 32(1):130-136, 1985. Google Scholar
  29. Hiroshi Imai and Takao Asano. Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. Journal of algorithms, 4(4):310-323, 1983. Google Scholar
  30. Klaus Jansen and Guochuan Zhang. On rectangle packing: maximizing benefits. In J. Ian Munro, editor, Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 204-213. SIAM, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.982822.
  31. Sanjeev Khanna, S. Muthukrishnan, and Mike Paterson. On approximating rectangle tiling and packing. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 384-393. ACM/SIAM, 1998. URL: http://dl.acm.org/citation.cfm?id=314613.314768.
  32. James T Klosowski, Martin Held, Joseph SB Mitchell, Henry Sowizral, and Karel Zikan. Efficient collision detection using bounding volume hierarchies of k-dops. IEEE transactions on Visualization and Computer Graphics, 4(1):21-36, 1998. Google Scholar
  33. Yoshiyuki Kusakari, Hitoshi Suzuki, and Takao Nishizeki. A shortest pair of paths on the plane with obstacles and crossing areas. International Journal of Computational Geometry & Applications, 9(02):151-170, 1999. Google Scholar
  34. Brian Lent, Arun Swami, and Jennifer Widom. Clustering association rules. In Proceedings 13th International Conference on Data Engineering, pages 220-231. IEEE, 1997. Google Scholar
  35. Ewa Malesinska. Graph theoretical models for frequency assignment problems. Citeseer, 1997. Google Scholar
  36. Dániel Marx. Efficient approximation schemes for geometric problems? In 13th Annual European Symposium on Algorithms (ESA), volume 3669, pages 448-459. Springer, 2005. URL: https://doi.org/10.1007/11561071_41.
  37. Joseph S. B. Mitchell. Approximating maximum independent set for rectangles in the plane. CoRR, abs/2101.00326, 2021. Version 1, URL: https://arxiv.org/abs/2101.00326v1.
  38. Joseph SB Mitchell. Approximating maximum independent set for rectangles in the plane. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 339-350. IEEE, 2022. Google Scholar
  39. Frank Nielsen. Fast stabbing of boxes in high dimensions. Theor. Comput. Sci., 246(1-2):53-72, 2000. URL: https://doi.org/10.1016/S0304-3975(98)00336-3.
  40. Bram Verweij and Karen Aardal. An optimisation algorithm for maximum independent set with applications in map labelling. In Algorithms-ESA’99: 7th Annual European Symposium Prague, Czech Republic, July 16-18, 1999 Proceedings 7, pages 426-437. Springer, 1999. Google Scholar
  41. Andreas Wiese. Independent set of convex polygons: From n^ε to 1+ε via shrinking. Algorithmica, 80(3):918-934, 2018. Google Scholar
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