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

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

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Keywords

  • Connectivity
  • Hamiltonian cycle
  • Polynomial space
  • Treedepth

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