Fast algorithms for solving the Hamilton Cycle problem with high probability

TL;DR

This paper introduces efficient algorithms that, with high probability, either find a Hamilton cycle or prove its absence in random graphs, with optimal running times depending on the graph representation.

Contribution

It provides the first deterministic algorithms with optimal running times for detecting Hamilton cycles in random graphs under different input formats.

Findings

Algorithms run in w.h.p. O(n) and O(n/p) time

Algorithms correctly identify Hamilton cycles or their absence

Optimal performance in both input settings

Abstract

We study the Hamilton cycle problem with input a random graph G=G(n,p) in two settings. In the first one, G is given to us in the form of randomly ordered adjacency lists while in the second one we are given the adjacency matrix of G. In each of the settings we give a deterministic algorithm that w.h.p. either it finds a Hamilton cycle or it returns a certificate that such a cycle does not exists, for p > 0. The running times of our algorithms are w.h.p. O(n) and O(n/p) respectively each being best possible in its own setting.