Deterministic “Snakes and Ladders” Heuristic for the Hamiltonian cycle problem

Abstract

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.

HybridHAM: A Novel Hybrid Heuristic for Finding Hamiltonian Cycle

“…Many TSP heuristics like the famous Lin-Kernighan heuristic [17,18] use a technique called "k-opt" transformation [19,20], which is an exchange of k edges. Baniasadi et al [21] proposed a "snakes and ladders" heuristic for solving HCP inspired from k-opt transformations. There are very few published HCP heuristics that sit in the "middle" area between sophisticated reliable heuristics and very simplistic (usually linear or quadratic time) approaches.…”

A note on using the resistance-distance matrix to solve Hamiltonian cycle problem

“…Although specialised algorithms and heuristics for solving HCP do exist (e.g. see Baniasadi et al (2014); Chalaturnyk (2008); Eppstein (2003)), a common and typically quite successful method for solving HCP is to first cast it as a TSP problem, and then use one of the suite of highly-developed TSP heuristics (e.g. see Applegate et al (2006); Helsgaun (2000)) to solve it.…”

A Linearly-Growing Conversion from the Set Splitting Problem to the Directed Hamiltonian Cycle Problem

“…Indeed, a famous instance of HCP -the so-called "Knight's tour" problem -was solved by Euler in the 1750s, and it remains an area of active research (e.g. see Eppstein [8], Borkar et al [3], and Baniasadi et al [2]).…”