eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-08-21
19:1
19:14
10.4230/LIPIcs.MFCS.2023.19
article
Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations
Biton, Noy
1
Levi, Reut
1
https://orcid.org/0000-0003-3167-1766
Medina, Moti
2
https://orcid.org/0000-0002-5572-3754
Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel
Faculty of Engineering, Bar-Ilan University, Ramat Gan, Israel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol272-mfcs2023/LIPIcs.MFCS.2023.19/LIPIcs.MFCS.2023.19.pdf
the CONGEST model
Hamiltonian Path
Hamiltonian Cycle
Dirac graphs
Ore graphs
graph-algorithms