The traveling salesman problem for cubic graphs

TL;DR

This paper presents improved algorithms for finding Hamiltonian cycles and solving the traveling salesman problem in cubic and degree-four graphs, achieving faster runtimes and enumeration capabilities.

Contribution

The paper introduces new algorithms with faster exponential runtimes for Hamiltonian cycle detection and TSP in cubic and degree-four graphs, including cycle counting and listing methods.

Findings

Algorithms run in time O(2^{n/3}) for cubic graphs.

Can count or list all Hamiltonian cycles in degree three graphs.

Provides bounds on the number of Hamiltonian cycles in degree three graphs.

Abstract

We show how to find a Hamiltonian cycle in a graph of degree at most three with n vertices, in time O(2^{n/3}) ~= 1.260^n and linear space. Our algorithm can find the minimum weight Hamiltonian cycle (traveling salesman problem), in the same time bound. We can also count or list all Hamiltonian cycles in a degree three graph in time O(2^{3n/8}) ~= 1.297^n. We also solve the traveling salesman problem in graphs of degree at most four, by randomized and deterministic algorithms with runtime O((27/4)^{n/3}) ~= 1.890^n and O((27/4+epsilon)^{n/3}) respectively. Our algorithms allow the input to specify a set of forced edges which must be part of any generated cycle. Our cycle listing algorithm shows that every degree three graph has O(2^{3n/8}) Hamiltonian cycles; we also exhibit a family of graphs with 2^{n/3} Hamiltonian cycles per graph.