Backtracking (the) Algorithms on the Hamiltonian Cycle Problem

Abstract

Even though the Hamiltonian cycle problem is NP-complete, many of its problem instances aren't. In fact, almost all the hard instances reside in one area: near the Koml\'os-Szemer\'edi bound, of $\frac{1}{2}\ v\cdot ln(v) + \frac{1}{2}\ v\cdot ln( ln(v))$ edges, where randomly generated graphs have an approximate 50\% chance of being Hamiltonian. If the number of edges is either much higher or much lower, the problem is not hard -- most backtracking algorithms decide such instances in (near) polynomial time. Recently however, targeted search efforts have identified very hard Hamiltonian cycle problem instances very far away from the Koml\'os-Szemer\'edi bound. In that study, the used backtracking algorithm was Vandegriend-Culberson's, which was supposedly the most efficient of all Hamiltonian backtracking algorithms. In this paper, we make a unified large scale quantitative comparison for the best known backtracking algorithms described between 1877 and 2016. We confirm the suspicion that the Koml\'os-Szemer\'edi bound is a hard area for all backtracking algorithms, but also that Vandegriend-Culberson is indeed the most efficient algorithm, when expressed in consumed computing time. When measured in recursive effectiveness however, the algorithm by Frank Rubin, almost half a century old, performs best. In a more general algorithmic assessment, we conjecture that edge pruning and non-Hamiltonicity checks might be largely responsible for these recursive savings. When expressed in system time however, denser problem instances require much more time per recursion. This is most likely due to the costliness of the extra search pruning procedures, which are relatively elaborate. We supply large amounts of experimental data, and a unified single-program implementation for all six algorithms. All data and algorithmic source code is made public for further use by our colleagues.