哈密顿圈问题的确定性“蛇梯”启发式算法
Deterministic "Snakes and Ladders" Heuristic for the Hamiltonian Cycle Problem
摘要：
本文提出了一个多项式复杂性的、确定性的、启发式的方法来求解$n$阶无向图中的哈密顿圈问题。虽然找到一个哈密顿圈在理论上是没有保证的，我们已经观察到，启发式是成功的，即使在这种情况下，循环是非常罕见的，它也表现得非常好，在TSPLIB网页上列出的大型图的所有HCP实例。启发式算法的名字来源于其迭代的可视化。图的所有顶点都以某种顺序放置在给定的圆上。图的边被分类为蛇或梯子，蛇形成圆的弧，梯子形成圆的弦。启发式算法努力将$n$蛇精确地放置在圆上，从而形成哈密顿圈。蛇和梯子启发式（SLH）使用受$k-$ opt算法（如现在经典的Lin-Kernighan启发式）启发的变换来重新排序圆上的顶点，以便将一些梯子转换为蛇，反之亦然。如果在$n^3$主要迭代中没有改进，则使用合适的停止准则确保启发式在多项式时间内终止。
英文摘要：
We present a polynomial complexity, deterministic, heuristic for solving the Hamiltonian Cycle Problem (HCP) in an undirected graph of order $n$. Although finding a Hamiltonian cycle is not theoretically guaranteed, we have observed that the heuristic is successful even in cases where such cycles are extremely rare, and it also performs very well on all HCP instances of large graphs listed on the TSPLIB web page. The heuristic owes its name to a visualisation of its iterations. All vertices of the graph are placed on a given circle in some order. The graph's edges are classified as either snakes or ladders, with snakes forming arcs of the circle and ladders forming its chords. The heuristic strives to place exactly $n$ snakes on the circle, thereby forming a Hamiltonian cycle. The Snakes and Ladders Heuristic (SLH) uses transformations inspired by $k-$opt algorithms such as the, now classical, Lin-Kernighan heuristic to reorder the vertices on the circle in order to transform some ladders into snakes and vice versa. The use of a suitable stopping criterion ensures the heuristic terminates in polynomial time if no improvement is made in $n^3$ major iterations.
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