Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

Jesper Nederlof, Michal Pilipczuk, Céline M. F. Swennenhuis, Karol Wegrzycki

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

13 Citations (Scopus)

Abstract

For many algorithmic problems on graphs of treewidth t , a standard dynamic programming approach gives an algorithm with time and space complexity 2O(t)⋅nO(1) . It turns out that when one considers the more restrictive parameter treedepth, it is often the case that a variation of this technique can be used to reduce the space complexity to polynomial, while retaining time complexity of the form 2O(d)⋅nO(1) , where d is the treedepth. This transfer of methodology is, however, far from automatic. For instance, for problems with connectivity constraints, standard dynamic programming techniques give algorithms with time and space complexity 2O(tlogt)⋅nO(1) on graphs of treewidth t , but it is not clear how to convert them into time-efficient polynomial space algorithms for graphs of low treedepth. Cygan et al. (FOCS’11) introduced the Cut&Count technique and showed that a certain class of problems with connectivity constraints can be solved in time and space complexity 2O(t)⋅nO(1) . Recently, Hegerfeld and Kratsch (STACS’20) showed that, for some of those problems, the Cut&Count technique can be also applied in the setting of treedepth, and it gives algorithms with running time 2O(d)⋅nO(1) and polynomial space usage. However, a number of important problems eluded such a treatment, with the most prominent examples being Hamiltonian Cycle and Longest Path. In this paper we clarify the situation by showing that Hamiltonian Cycle, Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover all admit 5d⋅nO(1) -time and polynomial space algorithms on graphs of treedepth d . The algorithms are randomized Monte Carlo with only false negatives.
Original languageEnglish
Title of host publicationGraph-Theoretic Concepts in Computer Science
Subtitle of host publication46th International Workshop, WG 2020, Leeds, UK, June 24–26, 2020, Revised Selected Papers
EditorsIsolde Adler, Haiko Müller
PublisherSpringer
Pages27-39
Number of pages13
ISBN (Electronic)978-3-030-60440-0
ISBN (Print)978-3-030-60439-4
DOIs
Publication statusPublished - 2020

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12301 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Funding

Jesper Nederlof is supported by the project CRACKNP that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 853234). Michał Pilipczuk is supported by the project TOTAL that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 677651). Céline M. F. Swennenhuis is supported by the Netherlands Organization for Scientific Research under project no. 613.009.031b. Karol Węgrzycki is supported by the grants 2016/21/N/ST6/01468 and 2018/28/T/ST6/00084 of the Polish National Science Center and project TOTAL that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 677651). Acknowledgements. We would like to thank anonymous reviewers and Petr Hliněný for their suggestions and comments. The second author would like to thank Marcin Wrochna for some early discussions on combining treedepth and Cut&Count. The work leading to the results presented in this paper was initiated during the Parameterized Retreat of the algorithms group of the University of Warsaw (PARUW), held in Karpacz in February 2019. This Retreat was financed by the project CUTACOMBS, which has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 714704).

FundersFunder number
Polish National Science Center
European Union's Horizon 2020 - Research and Innovation Framework Programme714704, 677651, 853234
European Research Council
Nederlandse Organisatie voor Wetenschappelijk Onderzoek613.009.031, 2016/21/N/ST6/01468, 2018/28/T/ST6/00084

    Keywords

    • Connectivity
    • Hamiltonian cycle
    • Polynomial space
    • Treedepth

    Fingerprint

    Dive into the research topics of 'Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space'. Together they form a unique fingerprint.

    Cite this