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Subexponential Algorithms for Clique Cover on Unit Disk and Unit Ball Graphs

Authors Tomohiro Koana , Nidhi Purohit , Kirill Simonov

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Author Details

Tomohiro Koana
  • Utrecht University, The Netherlands
Nidhi Purohit
  • National University of Singapore, Singapore
Kirill Simonov
  • Hasso Plattner Institute, University of Potsdam, Germany

Funding

  • Koana, Tomohiro: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (project CRACKNP under grant agreement No. 853234).
  • Simonov, Kirill: Supported by DFG Research Group ADYN via grant DFG 411362735.

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Tomohiro Koana, Nidhi Purohit, and Kirill Simonov. Subexponential Algorithms for Clique Cover on Unit Disk and Unit Ball Graphs. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.10

Abstract

In Clique Cover, given a graph G and an integer k, the task is to partition the vertices of G into k cliques. Clique Cover on unit ball graphs has a natural interpretation as a clustering problem, where the objective function is the maximum diameter of a cluster.
Many classical NP-hard problems are known to admit 2^{O(n^{1 - 1/d})}-time algorithms on unit ball graphs in ℝ^d [de Berg et al., SIAM J. Comp 2018]. A notable exception is the Maximum Clique problem, which admits a polynomial-time algorithm on unit disk graphs and a subexponential algorithm on unit ball graphs in ℝ³, but no subexponential algorithm on unit ball graphs in dimensions 4 or larger, assuming the ETH [Bonamy et al., JACM 2021].
In this work, we show that Clique Cover also suffers from a "curse of dimensionality", albeit in a significantly different way compared to Maximum Clique. We present a 2^{O(√n)}-time algorithm for unit disk graphs and argue that it is tight under the ETH. On the other hand, we show that Clique Cover does not admit a 2^{o(n)}-time algorithm on unit ball graphs in dimension 5, unless the ETH fails.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Clique cover
  • diameter clustering
  • subexponential algorithms
  • unit disk graphs

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