The Hardest Hamiltonian Cycle Problem Instances: The Plateau of Yes and the Cliff of No

Abstract

We use two evolutionary algorithms to make hard instances of the Hamiltonian cycle problem. Hardness (or ‘fitness’), is defined as the number of recursions required by Vandegriend–Culberson, the best known exact backtracking algorithm for the problem. The hardest instances, all non-Hamiltonian, display a high degree of regularity and scalability across graph sizes. These graphs are found multiple times through independent runs, and by both evolutionary algorithms, suggesting the search space might contain monotonic paths towards the global maximum. For Hamiltonian-bound evolution, some hard graphs were found, but convergence is much less consistent. In this extended paper, we survey the neighbourhoods of both the hardest yes- and no-instances produced by the evolutionary algorithms. Results show that the hardest no-instance resides on top of a steep cliff, while the hardest yes-instance turns out to be part of a plateau of 27 equally hard instances. While definitive answers are far away, the results provide a lot of insight in the Hamiltonian cycle problem’s state space.

Hamiltonian cycles on Ammann-Beenker Tilings

“…In the context of the NPC problem, exact solutions to Hamiltonian cycles are useful because they serve as bench-marking measures for new algorithms [91][92][93]. While the difficulty of NPC means no known algorithm is capable of deterministically solving all instances in polynomial time, several algorithms display a high success rate on many graphs [91,94,95].…”