Finding a Lower Bound for k-Unbounded Hamiltonian Cycles

TL;DR

This paper introduces the concept of m-Unbounded Hamiltonian Cycles, explores their existence in non-Hamiltonian graphs, and proposes algorithms including a heuristic for finding such cycles.

Contribution

It defines a new variant of the Hamiltonian Cycle Problem, provides an exponential brute-force algorithm, and develops a polynomial-time heuristic for practical solutions.

Findings

Exponential-time brute-force algorithm for m-Unbounded Hamiltonian Cycles.

Transformations to Hamiltonian Cycle and TSP problems for solving the variant.

A polynomial-time heuristic effectively approximates solutions.

Abstract

Methods to determine the existence of Hamiltonian Cycles in graphs have been extensively studied. However, little research has been done following cases when no Hamiltonian Cycle exists. Let a vertex be "unbounded" if it is visited more than once in a path. Furthermore, let a k-Unbounded Hamiltonian Cycle be a path with finite length that visits every vertex, has adjacent start and end vertices, and contains k unbounded vertices. We consider a novel variant of the Hamiltonian Cycle Problem in which the objective is to find an m-Unbounded Hamiltonian Cycle where m is the minimum value of k such that a k-Unbounded Hamiltonian Cycle exists. We first consider the task on well-known non-Hamiltonian graphs. We then provide an exponential-time brute-force algorithm for the determination of an m-Unbounded Hamiltonian Cycle and discuss approaches to solve the variant through transformations to the Hamiltonian Cycle Problem and the Asymmetric Traveling Salesman Problem. Finally, we present a polynomial-time heuristic for the determination of an m-Unbounded Hamiltonian Cycle that is also shown to be an effective heuristic for the original Hamiltonian Cycle Problem.