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On the Exact Complexity of Hamiltonian Cycle and q-Colouring in Disk Graphs

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Algorithms and Complexity (CIAC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10236))

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Abstract

We study the exact complexity of the Hamiltonian Cycle and the q-Colouring problem in disk graphs. We show that the Hamiltonian Cycle problem can be solved in \(2^{O(\sqrt{n})}\) on n-vertex disk graphs where the ratio of the largest and smallest disk radius is O(1). We also show that this is optimal: assuming the Exponential Time Hypothesis, there is no \(2^{o(\sqrt{n})}\)-time algorithm for Hamiltonian Cycle, even on unit disk graphs. We give analogous results for graph colouring: under the Exponential Time Hypothesis, for any fixed q, q-Colouring does not admit a \(2^{o(\sqrt{n})}\)-time algorithm, even when restricted to unit disk graphs, and it is solvable in \(2^{O(\sqrt{n})}\)-time on disk graphs.

The research of the first author was supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 024.002.003.

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Acknowledgements

This work was initiated at the Lorentz Center workshop ‘Fixed-Parameter Computational Geometry’. We are grateful to Hans L. Bodlaender and Mark de Berg for discussions and their help with improving this paper.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, TU Eindhoven, Eindhoven, The Netherlands

    Sándor Kisfaludi-Bak

  2. Department of Computer Science, Utrecht University, Utrecht, The Netherlands

    Tom C. van der Zanden

Authors
  1. Sándor Kisfaludi-Bak
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  2. Tom C. van der Zanden
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Correspondence to Sándor Kisfaludi-Bak .

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Editors and Affiliations

  1. National Technical University of Athens, Zografou, Athens, Greece

    Dimitris Fotakis

  2. National Technical University of Athens, Zografou, Athens, Greece

    Aris Pagourtzis

  3. Université Paris-Dauphine, Paris, France

    Vangelis Th. Paschos

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Kisfaludi-Bak, S., van der Zanden, T.C. (2017). On the Exact Complexity of Hamiltonian Cycle and q-Colouring in Disk Graphs. In: Fotakis, D., Pagourtzis, A., Paschos, V. (eds) Algorithms and Complexity. CIAC 2017. Lecture Notes in Computer Science(), vol 10236. Springer, Cham. https://doi.org/10.1007/978-3-319-57586-5_31

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  • DOI: https://doi.org/10.1007/978-3-319-57586-5_31

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  • Online ISBN: 978-3-319-57586-5

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