License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2023.19
URN: urn:nbn:de:0030-drops-185534
URL: https://drops.dagstuhl.de/opus/volltexte/2023/18553/
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Biton, Noy ; Levi, Reut ; Medina, Moti

Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations

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LIPIcs-MFCS-2023-19.pdf (0.9 MB)


Abstract

We study the problem of finding a Hamiltonian cycle under the promise that the input graph has a minimum degree of at least n/2, where n denotes the number of vertices in the graph. The classical theorem of Dirac states that such graphs (a.k.a. Dirac graphs) are Hamiltonian, i.e., contain a Hamiltonian cycle. Moreover, finding a Hamiltonian cycle in Dirac graphs can be done in polynomial time in the classical centralized model.
This paper presents a randomized distributed CONGEST algorithm that finds w.h.p. a Hamiltonian cycle (as well as maximum matching) within O(log n) rounds under the promise that the input graph is a Dirac graph. This upper bound is in contrast to general graphs in which both the decision and search variants of Hamiltonicity require Ω̃(n²) rounds, as shown by Bachrach et al. [PODC'19].
In addition, we consider two generalizations of Dirac graphs: Ore graphs and Rahman-Kaykobad graphs [IPL'05]. In Ore graphs, the sum of the degrees of every pair of non-adjacent vertices is at least n, and in Rahman-Kaykobad graphs, the sum of the degrees of every pair of non-adjacent vertices plus their distance is at least n+1. We show how our algorithm for Dirac graphs can be adapted to work for these more general families of graphs.

BibTeX - Entry

@InProceedings{biton_et_al:LIPIcs.MFCS.2023.19,
  author =	{Biton, Noy and Levi, Reut and Medina, Moti},
  title =	{{Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{19:1--19:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18553},
  URN =		{urn:nbn:de:0030-drops-185534},
  doi =		{10.4230/LIPIcs.MFCS.2023.19},
  annote =	{Keywords: the CONGEST model, Hamiltonian Path, Hamiltonian Cycle, Dirac graphs, Ore graphs, graph-algorithms}
}

Keywords: the CONGEST model, Hamiltonian Path, Hamiltonian Cycle, Dirac graphs, Ore graphs, graph-algorithms
Collection: 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)
Issue Date: 2023
Date of publication: 21.08.2023


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