An Optimal XP Algorithm for Hamiltonian Cycle on Graphs of Bounded Clique-Width

Abstract

In this paper, we prove that, given a clique-width k-expression of an n-vertex graph, Hamiltonian Cycle can be solved in time \(n^{\mathcal {O}(k)}\). This improves the naive algorithm that runs in time \(n^{\mathcal {O}(k^2)}\) by Espelage et al. (Graph-theoretic concepts in computer science, vol 2204. Springer, Berlin, 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time \(n^{o(k)}\) (Fomin et al. in SIAM J Comput 43:1541–1563, 2014). We present a technique of representative sets using two-edge colored multigraphs on k vertices. The essential idea is that, for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternately can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on k vertices with at most n edges.

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Acknowledgements

The authors would like to thank the anonymous reviewer for pointing out the previous mistake on red–blue Eulerian trails for directed graphs and for indicating proper citations.

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Correspondence to O-joung Kwon.

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An extended abstract appeared in Algorithms and Data Structures, WADS 2017 [3]. B. Bergougnoux and M.M. Kanté are supported by French Agency for Research under the GraphEN Project (ANR-15-CE40-0009). O. Kwon is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC consolidator grant DISTRUCT, Agreement No. 648527), also supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education (No. NRF-2018R1D1A1B07050294), and also supported by Institute for Basic Sciences (IBS-R029-C1).

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Bergougnoux, B., Kanté, M.M. & Kwon, Oj. An Optimal XP Algorithm for Hamiltonian Cycle on Graphs of Bounded Clique-Width. Algorithmica 82, 1654–1674 (2020). https://doi.org/10.1007/s00453-019-00663-9

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Keywords

  • Hamiltonian cycle
  • Eulerian trail
  • Clique-width
  • XP algorithm