Maximum Clique in Disk-Like Intersection Graphs

Authors Édouard Bonnet , Nicolas Grelier, Tillmann Miltzow

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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

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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


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