The Gn,m Phase Transition is Not Hard for the Hamiltonian Cycle Problem

TL;DR

This study demonstrates that the Hamiltonian Cycle problem remains computationally easy across various graph classes, challenging the notion that the Gn,m phase transition correlates with problem hardness.

Contribution

The paper introduces an improved backtrack algorithm and provides evidence that the Gn,m phase transition does not lead to a high frequency of hard instances in the Hamiltonian Cycle problem.

Findings

All tested graphs of 100 to 1500 vertices are easily solved.

Restricting degree sequences still results in low frequency of hard graphs.

Certain regular graphs can be very hard, but with bounded average degree.

Abstract

Using an improved backtrack algorithm with sophisticated pruning techniques, we revise previous observations correlating a high frequency of hard to solve Hamiltonian Cycle instances with the Gn,m phase transition between Hamiltonicity and non-Hamiltonicity. Instead all tested graphs of 100 to 1500 vertices are easily solved. When we artificially restrict the degree sequence with a bounded maximum degree, although there is some increase in difficulty, the frequency of hard graphs is still low. When we consider more regular graphs based on a generalization of knight's tours, we observe frequent instances of really hard graphs, but on these the average degree is bounded by a constant. We design a set of graphs with a feature our algorithm is unable to detect and so are very hard for our algorithm, but in these we can vary the average degree from O(1) to O(n). We have so far found no class of graphs correlated with the Gn,m phase transition which asymptotically produces a high frequency of hard instances.