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An Asymptotically Fast Polynomial Space Algorithm for Hamiltonicity Detection in Sparse Directed Graphs

Author Andreas Björklund

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Andreas Björklund
  • Private, Lund, Sweden - This work was done while being employed by Ericsson Research

Acknowledgements

We are very grateful to an anonymous reviewer who pointed out a serious flaw in an earlier proof attempt of Lemma 6. We also thank Łukasz Kowalik for comments on an earlier version of the paper.

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Andreas Björklund. An Asymptotically Fast Polynomial Space Algorithm for Hamiltonicity Detection in Sparse Directed Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 15:1-15:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.STACS.2021.15

Abstract

We present a polynomial space Monte Carlo algorithm that given a directed graph on n vertices and average outdegree δ, detects if the graph has a Hamiltonian cycle in 2^{n-Ω(n/δ)} time. This asymptotic scaling of the savings in the running time matches the fastest known exponential space algorithm by Björklund and Williams ICALP 2019. By comparison, the previously best polynomial space algorithm by Kowalik and Majewski IPEC 2020 guarantees a 2^{n-Ω(n/(2^δ))} time bound.
Our algorithm combines for the first time the idea of obtaining a fingerprint of the presence of a Hamiltonian cycle through an inclusion-exclusion summation over the Laplacian of the graph from Björklund, Kaski, and Koutis ICALP 2017, with the idea of sieving for the non-zero terms in an inclusion-exclusion summation by listing solutions to systems of linear equations over ℤ₂ from Björklund and Husfeldt FOCS 2013.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Hamiltonian cycle
  • directed graph
  • worst case analysis algorithm

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References

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