# Deterministic "Snakes and Ladders" Heuristic for the Hamiltonian Cycle Problem

## TL;DR

This paper introduces a polynomial-time deterministic heuristic called the Snakes and Ladders Heuristic (SLH) for approximating Hamiltonian cycles in undirected graphs, performing well even on challenging instances.

## Contribution

The paper presents a novel polynomial complexity heuristic inspired by $k$-opt algorithms, capable of finding Hamiltonian cycles or near-solutions in large graphs.

## Key findings

- Successful in finding Hamiltonian cycles in large TSPLIB graphs
- Performs well even when Hamiltonian cycles are rare
- Operates within polynomial time with a suitable stopping criterion

## 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.