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.
翻译:尽管汉密尔顿周期问题已经完全解决,但许多问题却不是。事实上,几乎所有困难事件都存在于一个领域:接近 Kolml\'os-Szemer\'edi 约束,即$\frac{1 ⁇ 2\\\ v\cdot In(v) +\frac{1 ⁇ 2\\\\ v\cdot In(in(v)) 边缘,其中随机生成的图表有大约50 ⁇ 机会成为汉密尔顿式。如果边缘数要么高得多,要么低得多,问题就不难了。如果大多数反跟踪算法在(近)多元时间里决定这类情况。但最近,有针对性地搜索发现汉密尔顿周期周期存在非常困难的问题非常远远。在研究中,所使用的回溯算算法是范德格林-Culberson 更深入的时期,这似乎是所有汉密尔密尔顿的半轨算方法的效率。在本文中,我们做了一个规模上最大规模的定量的数值比较,在18个已知的时期里,我们可能要确认一个已知的旧算。