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

TL;DR

This paper introduces a fast randomized distributed algorithm in the CONGEST model that efficiently finds Hamiltonian cycles in Dirac graphs and their generalizations, significantly improving over general graph algorithms.

Contribution

It presents the first sublogarithmic-round distributed algorithm for Hamiltonian cycle detection in Dirac graphs and extends it to Ore and Rahman-Kaykobad graphs.

Findings

Runs in O(log n) rounds with high probability

Efficiently finds Hamiltonian cycles in Dirac graphs

Adapts to Ore and Rahman-Kaykobad graph classes

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 $\tilde{\Omega}(n^2)$ 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.