Document Open Access Logo

Maximum Clique in Disk-Like Intersection Graphs

Authors Édouard Bonnet , Nicolas Grelier, Tillmann Miltzow

Thumbnail PDF


  • Filesize: 0.69 MB
  • 18 pages

Document Identifiers

Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Nicolas Grelier
  • Department of Computer Science, ETH Zürich, Switzerland
Tillmann Miltzow
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Cite AsGet BibTex

Édouard Bonnet, Nicolas Grelier, and Tillmann Miltzow. Maximum Clique in Disk-Like Intersection Graphs. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 17:1-17:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. '18, Bonamy et al. '18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard. We observe that, so far, all the hardness proofs for Maximum Clique in intersection graph classes I follow the same road. They show that, for every graph G of a large-enough class C, the complement of an even subdivision of G belongs to the intersection class I. Then they conclude by invoking the hardness of Maximum Independent Set on the class C, and the fact that the even subdivision preserves that hardness. However there is a strong evidence that this approach cannot work for disk graphs [Bonnet et al. '18]. We suggest a new approach, based on a problem that we dub Max Interval Permutation Avoidance, which we prove unlikely to have a subexponential-time approximation scheme. We transfer that hardness to Maximum Clique in intersection graphs of objects which can be either half-planes (or unit disks) or axis-parallel rectangles. That problem is not amenable to the previous approach. We hope that a scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance could help making progress on the disk case, for instance by showing the NP-hardness for (convex) pseudo-disks.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph algorithms
  • Disk Graphs
  • Intersection Graphs
  • Maximum Clique
  • Algorithms
  • NP-hardness
  • APX-hardness


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


  1. Anders Aamand, Mikkel Abrahamsen, Jakob B. T. Knudsen, and Peter M. R. Rasmussen. Classifying convex bodies by their contact and intersection graphs. CoRR, abs/1902.01732, 2019. URL:
  2. Akanksha Agrawal, Grzegorz Guspiel, Jayakrishnan Madathil, Saket Saurabh, and Meirav Zehavi. Connecting the dots (with minimum crossings). In 35th International Symposium on Computational Geometry, SoCG 2019, June 18-21, 2019, Portland, Oregon, USA., pages 7:1-7:17, 2019. URL:
  3. Paola Alimonti and Viggo Kann. Some APX-completeness results for cubic graphs. Theor. Comput. Sci., 237(1-2):123-134, 2000. URL:
  4. Boris Aronov, Anirudh Donakonda, Esther Ezra, and Rom Pinchasi. On pseudo-disk hypergraphs. arXiv preprint arXiv:1802.08799, 2018. Google Scholar
  5. Piotr Berman and Marek Karpinski. Efficient amplifiers and bounded degree optimization. Electronic Colloquium on Computational Complexity (ECCC), 8(53), 2001. URL:
  6. Marthe Bonamy, Édouard Bonnet, Nicolas Bousquet, Pierre Charbit, and Stéphan Thomassé. EPTAS for max clique on disks and unit balls. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 568-579, 2018. URL:
  7. Édouard Bonnet, Panos Giannopoulos, Eun Jung Kim, Paweł Rzążewski, and Florian Sikora. QPTAS and subexponential algorithm for maximum clique on disk graphs. In 34th International Symposium on Computational Geometry, SoCG 2018, June 11-14, 2018, Budapest, Hungary, pages 12:1-12:15, 2018. URL:
  8. Édouard Bonnet, Nicolas Grelier, and Tillmann Miltzow. Maximum clique in disk-like intersection graphs. arXiv preprint arXiv:2003.02583, 2020. Google Scholar
  9. Édouard Bonnet and Tillmann Miltzow. Parameterized hardness of art gallery problems. In 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pages 19:1-19:17, 2016. URL:
  10. Édouard Bonnet and Paweł Rzążewski. Optimality program in segment and string graphs. Algorithmica, 81(7):3047-3073, 2019. URL:
  11. Andreas Brandstädt, Van Bang Le, and Jeremy P. Spinrad. Graph classes: a survey. SIAM, 1999. Google Scholar
  12. Heinz Breu and David G. Kirkpatrick. Unit disk graph recognition is NP-hard. Comput. Geom., 9(1-2):3-24, 1998. URL:
  13. Valentin E. Brimkov, Konstanty Junosza-Szaniawski, Sean Kafer, Jan Kratochvíl, Martin Pergel, Paweł Rzążewski, Matthew Szczepankiewicz, and Joshua Terhaar. Homothetic polygons and beyond: Maximal cliques in intersection graphs. Discrete Applied Mathematics, 247:263-277, 2018. URL:
  14. Ayelet Butman, Danny Hermelin, Moshe Lewenstein, and Dror Rawitz. Optimization problems in multiple-interval graphs. ACM Trans. Algorithms, 6(2):40:1-40:18, 2010. URL:
  15. Sergio Cabello, Jean Cardinal, and Stefan Langerman. The clique problem in ray intersection graphs. Discrete & Computational Geometry, 50(3):771-783, 2013. URL:
  16. Paz Carmi, Matthew J. Katz, and Pat Morin. Stabbing pairwise intersecting disks by four points. CoRR, abs/1812.06907, 2018. URL:
  17. Jérémie Chalopin and Daniel Gonçalves. Every planar graph is the intersection graph of segments in the plane: extended abstract. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 631-638, 2009. URL:
  18. Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms, 46(2):178-189, 2003. URL:
  19. Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discrete Mathematics, 86(1-3):165-177, 1990. URL:
  20. Ludwig Danzer. Zur lösung des gallaischen problems über kreisscheiben in der euklidischen ebene. Studia Sci. Math. Hungar, 21(1-2):111-134, 1986. Google Scholar
  21. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  22. Y. G. Dorfman and G. I. Orlova. Finding the maximal cut in a graph. Engineering Cybernetics, 10:502-506, 1972. Google Scholar
  23. Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput., 34(6):1302-1323, 2005. URL:
  24. Aleksei V. Fishkin. Disk graphs: A short survey. In Approximation and Online Algorithms, First International Workshop, WAOA 2003, Budapest, Hungary, September 16-18, 2003, Revised Papers, pages 260-264, 2003. URL:
  25. Mathew C. Francis, Daniel Gonçalves, and Pascal Ochem. The Maximum Clique Problem in Multiple Interval Graphs. Algorithmica, 71(4):812-836, 2015. URL:
  26. Grzegorz Guspiel. Complexity of finding perfect bipartite matchings minimizing the number of intersecting edges. CoRR, abs/1709.06805, 2017. URL:
  27. Sariel Har-Peled, Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir, and Max Willert. Stabbing pairwise intersecting disks by five points. In 29th International Symposium on Algorithms and Computation, ISAAC 2018, December 16-19, 2018, Jiaoxi, Yilan, Taiwan, pages 50:1-50:12, 2018. URL:
  28. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. URL:
  29. Petr Hlinený and Jan Kratochvíl. Representing graphs by disks and balls (a survey of recognition-complexity results). Discrete Mathematics, 229(1-3):101-124, 2001. URL:
  30. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL:
  31. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, December 2001. Google Scholar
  32. Ross J. Kang and Tobias Müller. Sphere and Dot Product Representations of Graphs. Discrete & Computational Geometry, 47(3):548-568, 2012. URL:
  33. Seog-Jin Kim, Alexandr Kostochka, and Kittikorn Nakprasit. On the chromatic number of intersection graphs of convex sets in the plane. the electronic journal of combinatorics, 11(1):52, 2004. Google Scholar
  34. Paul Koebe. Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physikalische Klasse, 88:141-164, 1936. Google Scholar
  35. Jan Kratochvíl. String graphs. II. recognizing string graphs is NP-hard. J. Comb. Theory, Ser. B, 52(1):67-78, 1991. URL:
  36. Jan Kratochvíl and Jiří Matoušek. Intersection graphs of segments. J. Comb. Theory, Ser. B, 62(2):289-315, 1994. URL:
  37. Dániel Marx. Parameterized complexity of independence and domination on geometric graphs. In Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, pages 154-165, 2006. URL:
  38. Dániel Marx and Michal Pilipczuk. Optimal parameterized algorithms for planar facility location problems using voronoi diagrams. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 865-877, 2015. URL:
  39. Terry A. McKee and Fred R. McMorris. Topics in intersection graph theory. SIAM, 1999. Google Scholar
  40. Matthias Middendorf and Frank Pfeiffer. The max clique problem in classes of string-graphs. Discrete Mathematics, 108(1-3):365-372, 1992. URL:
  41. Bernard M. E. Moret. Planar NAE3SAT is in P. ACM SIGACT News, 19(2):51-54, 1988. Google Scholar
  42. Dana Moshkovitz and Ran Raz. Sub-constant error probabilistically checkable proof of almost-linear size. Computational Complexity, 19(3):367-422, 2010. URL:
  43. Tim Nieberg and Johann Hurink. A PTAS for the minimum dominating set problem in unit disk graphs. In Approximation and Online Algorithms, Third International Workshop, WAOA 2005, Palma de Mallorca, Spain, October 6-7, 2005, Revised Papers, pages 296-306, 2005. URL:
  44. Tim Nieberg, Johann Hurink, and Walter Kern. A robust PTAS for maximum weight independent sets in unit disk graphs. In Graph-Theoretic Concepts in Computer Science, 30th International Workshop, WG 2004, Bad Honnef, Germany, June 21-23, 2004, Revised Papers, pages 214-221, 2004. URL:
  45. Svatopluk Poljak. A note on stable sets and colorings of graphs. Commentationes Mathematicae Universitatis Carolinae, 15(2):307-309, 1974. Google Scholar
  46. Vijay Raghavan and Jeremy P. Spinrad. Robust algorithms for restricted domains. J. Algorithms, 48(1):160-172, 2003. URL:
  47. Marcus Schaefer, Eric Sedgwick, and Daniel Stefankovic. Recognizing string graphs in NP. J. Comput. Syst. Sci., 67(2):365-380, 2003. URL:
  48. Lajos Stachó. A solution of gallai’s problem on pinning down circles. Mat. Lapok, 32(1-3):19-47, 1981. Google Scholar
  49. Erik Jan van Leeuwen. Better approximation schemes for disk graphs. In Algorithm Theory - SWAT 2006, 10th ScandinavianWorkshop on Algorithm Theory, Riga, Latvia, July 6-8, 2006, Proceedings, pages 316-327, 2006. URL:
  50. Erik Jan van Leeuwen. Optimization and Approximation on Systems of Geometric Objects. PhD thesis, Utrecht University, 2009. 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