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Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs

Authors Jie Gao , Paweł Gawrychowski , Panos Giannopoulos , Wolfgang Mulzer , Satyam Singh , Frank Staals , Meirav Zehavi

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Maximum Clique
  • Disk Graphs
  • Unit Disk Graphs
  • FPT Approximation

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Abstract

A disk graph is the intersection graph of (closed) disks in the plane. We consider the classic problem of finding a maximum clique in a disk graph. For general disk graphs, the complexity of this problem is still open, but for unit disk graphs, it is well known to be in P. The currently fastest algorithm runs in time O(n^{7/3+ o(1)}), where n denotes the number of disks [Jared Espenant et al., 2023; J. Mark Keil and Debajyoti Mondal, 2025]. Moreover, for the case of disk graphs with t distinct radii, the problem has also recently been shown to be in XP. More specifically, it is solvable in time O^*(n^{2t}) [J. Mark Keil and Debajyoti Mondal, 2025]. In this paper, we present algorithms with improved running times by allowing for approximate solutions and by using randomization: [(i)] 
1) for unit disk graphs, we give an algorithm that, with constant success probability, computes a (1-ε)-approximate maximum clique in expected time Õ(n/ε²); and
2) for disk graphs with t distinct radii, we give a parameterized approximation scheme that, with a constant success probability, computes a (1-ε)-approximate maximum clique in expected time Õ(f(t)⋅ (1/ε)^{O(t)} ⋅ n), for some (exponential) function f(t).

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Jie Gao, Paweł Gawrychowski, Panos Giannopoulos, Wolfgang Mulzer, Satyam Singh, Frank Staals, and Meirav Zehavi. Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SWAT.2026.20

Author Details

Jie Gao
  • Computer Science Department, Rutgers University, Piscataway, NJ, USA
Paweł Gawrychowski
  • Institute of Computer Science, University of Wrocław, Poland
Panos Giannopoulos
  • Department of Computer Science, City St George’s, University of London, UK
Wolfgang Mulzer
  • Institut für Informatik, Freie Universität Berlin, Germany
Satyam Singh
  • Department of Computer Science, Aalto University, Finland
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Meirav Zehavi
  • Institute for the Theory of Computing, Ben Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Gao, Jie: Research on this paper was supported by NSF under grants CNS-2515159, IIS-2229876, DMS-2220271, DMS-2311064, and CCF-2118953.
  • Gawrychowski, Paweł: Research on this paper was supported by the Polish National Science Centre under grant 2023/51/B/ST6/01505.
  • Singh, Satyam: Research on this paper was supported by the Research Council of Finland, Grant 363444.
  • Zehavi, Meirav: Research on this paper was supported by he Israel Science Foundation under grant 1470/24, and European Research Council under grant 101039913 (PARAPATH).

Acknowledgements

The work in this paper was initiated at the Lorentz Center workshop on Fine-Grained & Parameterized Computational Geometry, held in February 2025.

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