Maximum Matchings in Geometric Intersection Graphs

Authors Édouard Bonnet , Sergio Cabello , Wolfgang Mulzer

Thumbnail PDF


  • Filesize: 0.57 MB
  • 17 pages

Document Identifiers

Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Sergio Cabello
  • Faculty of Mathematics and Physics, University of Ljubljana, and IMFM, Slovenia
Wolfgang Mulzer
  • Institut für Informatik, Freie Universität Berlin, Germany

Cite AsGet BibTex

Édouard Bonnet, Sergio Cabello, and Wolfgang Mulzer. Maximum Matchings in Geometric Intersection Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O(ρ^{3ω/2}n^{ω/2}) time with high probability, where ρ is the density of the geometric objects and ω>2 is a constant such that n × n matrices can be multiplied in O(n^ω) time. The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O(n^{ω/2}) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [1, Ψ] can be found in O(Ψ⁶log^11 n + Ψ^{12 ω} n^{ω/2}) time with high probability.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Graph algorithms analysis
  • computational geometry
  • geometric intersection graph
  • maximum matching
  • disk graph
  • unit-disk graph


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


  1. Pankaj. K. Agarwal, Sariel. Har-Peled, and Kasturi R. Varadarajan. Approximating extent measures of points. J. ACM, 51(4):606-635, 2004. URL:
  2. Pankaj K. Agarwal and Jiří Matoušek. Ray shooting and parametric search. SIAM J. Comput., 22(4):794-806, 1993. URL:
  3. Jochen Alber and Jirí Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms, 52(2):134-151, 2004. Google Scholar
  4. Noga Alon and Raphael Yuster. Matrix sparsification and nested dissection over arbitrary fields. J. ACM, 60(4):25:1-25:18, 2013. URL:
  5. douard Bonnet, Sergio Cabello, and Wolfgang Mulzer. Maximum matchings in geometric intersection graphs. CoRR, abs/1910.02123, 2019. URL:
  6. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational geometry: algorithms and applications, 3rd Edition. Springer, 2008. URL:
  7. Mark de Berg, A. Frank van der Stappen, Jules Vleugels, and Matthew J. Katz. Realistic input models for geometric algorithms. Algorithmica, 34(1):81-97, 2002. URL:
  8. Hristo Djidjev and Shankar M. Venkatesan. Reduced constants for simple cycle graph separation. Acta Inf., 34(3):231-243, 1997. Google Scholar
  9. Alon Efrat, Alon Itai, and Matthew J. Katz. Geometry helps in bottleneck matching and related problems. Algorithmica, 31(1):1-28, 2001. URL:
  10. David Eppstein, Gary L. Miller, and Shang-Hua Teng. A deterministic linear time algorithm for geometric separators and its applications. Fundam. Inform., 22(4):309-329, 1995. Google Scholar
  11. J. Gao and L. Guibas. Geometric algorithms for sensor networks. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 370(1958):27-51, 2011. URL:
  12. John R. Gilbert and Robert E. Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50(4):377-404, 1986. URL:
  13. Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. SIAM J. Comput., 46(6):1712-1744, 2017. URL:
  14. M. L. Huson and A. Sen. Broadcast scheduling algorithms for radio networks. In IEEE MILCOM '95, volume 2, pages 647-651 vol.2, 1995. URL:
  15. Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar voronoi diagrams for general distance functions and their algorithmic applications. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, pages 2495-2504. SIAM, 2017. URL:
  16. Santosh Kumar, Ten H. Lai, and Anish Arora. Barrier coverage with wireless sensors. Wireless Networks, 13(6):817-834, 2007. Google Scholar
  17. Richard J. Lipton, Donald J. Rose, and Robert E. Tarjan. Generalized nested dissection. SIAM J. Numer. Anal., 16(2):346-358, 1979. URL:
  18. Richard J. Lipton and Robert Endre Tarjan. Applications of a planar separator theorem. SIAM J. Comput., 9(3):615-627, 1980. Google Scholar
  19. Aleksander Mądry. Navigating central path with electrical flows: From flows to matchings, and back. In Proceedings of the 54th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2013, pages 253-262, 2013. URL:
  20. Silvio Micali and Vijay V. Vazirani. An O(√|V|⋅ |E|) algorithm for finding maximum matching in general graphs. In Proceedings of the 21st Annual Symposium on Foundations of Computer Science, FOCS 1980, pages 17-27. IEEE Computer Society, 1980. URL:
  21. Gary L. Miller, Shang-Hua Teng, William P. Thurston, and Stephen A. Vavasis. Separators for sphere-packings and nearest neighbor graphs. J. ACM, 44(1):1-29, 1997. Google Scholar
  22. Marcin Mucha and Piotr Sankowski. Maximum matchings via gaussian elimination. In Proceedings of the 45th Symposium on Foundations of Computer Science, FOCS 2004, pages 248-255. IEEE Computer Society, 2004. URL:
  23. Marcin Mucha and Piotr Sankowski. Maximum matchings in planar graphs via gaussian elimination. Algorithmica, 45(1):3-20, 2006. URL:
  24. Michael O. Rabin and Vijay V. Vazirani. Maximum matchings in general graphs through randomization. J. Algorithms, 10(4):557-567, 1989. URL:
  25. Warren D. Smith and Nicholas C. Wormald. Geometric separator theorems and applications. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science, FOCS, pages 232-243, 1998. Google Scholar
  26. A. Frank van der Stappen, Dan Halperin, and Mark H. Overmars. The complexity of the free space for a robot moving amidst fat obstacles. Comput. Geom., 3:353-373, 1993. URL:
  27. Vijay V. Vazirani. A simplification of the MV matching algorithm and its proof. CoRR, abs/1210.4594, 2012. URL:
  28. Raphael Yuster and Uri Zwick. Maximum matching in graphs with an excluded minor. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pages 108-117. SIAM, 2007. URL:
  29. F. Zhao and L. Guibas. Wireless Sensor Networks: An Information Processing Approach. Elsevier/Morgan-Kaufmann, 2004. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail