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Finding Shortest Paths Between Graph Colourings

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Abstract

The \(k\)-colouring reconfiguration problem asks whether, for a given graph \(G\), two proper \(k\)-colourings \(\alpha \) and \(\beta \) of \(G\), and a positive integer \(\ell \), there exists a sequence of at most \(\ell +1\) proper \(k\)-colourings of \(G\) which starts with \(\alpha \) and ends with \(\beta \) and where successive colourings in the sequence differ on exactly one vertex of \(G\). We give a complete picture of the parameterized complexity of the \(k\)-colouring reconfiguration problem for each fixed \(k\) when parameterized by \(\ell \). First we show that the \(k\)-colouring reconfiguration problem is polynomial-time solvable for \(k=3\), settling an open problem of Cereceda, van den Heuvel and Johnson. Then, for all \(k \ge 4\), we show that the \(k\)-colouring reconfiguration problem, when parameterized by \(\ell \), is fixed-parameter tractable (addressing a question of Mouawad, Nishimura, Raman, Simjour and Suzuki) but that it has no polynomial kernel unless the polynomial hierarchy collapses.

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Notes

  1. PSPACE-completeness appears to be the default complexity for intractable instances of this kind of problem; see [16].

  2. A (polynomial) compression is a relaxed form of (polynomial) kernelization: The output may be with respect to any (possibly unparameterized) problem.

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Acknowledgments

We are grateful to several reviewers for insightful comments that greatly improved our presentation.

Author information

Author notes

  1. Viresh Patel

    Present address: University of Amsterdam, P.O. Box 19268, 1000 GG, Amsterdam, The Netherlands

Authors and Affiliations

  1. School of Engineering and Computing Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, UK

    Matthew Johnson & Daniël Paulusma

  2. Laboratoire d’Informatique Théorique et Appliquée, Université de Lorraine, 57045, Metz Cedex 01, France

    Dieter Kratsch

  3. Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Berlin, Germany

    Stefan Kratsch

  4. School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK

    Viresh Patel

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  1. Matthew Johnson
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  2. Dieter Kratsch
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  5. Daniël Paulusma
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Corresponding author

Correspondence to Matthew Johnson.

Additional information

This study was supported by EPSRC (EP/G043434/1), by a Scheme 7 Grant from the London Mathematical Society, and by the German Research Foundation (KR 4286/1).

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Johnson, M., Kratsch, D., Kratsch, S. et al. Finding Shortest Paths Between Graph Colourings. Algorithmica 75, 295–321 (2016). https://doi.org/10.1007/s00453-015-0009-7

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  • DOI: https://doi.org/10.1007/s00453-015-0009-7

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