A New Upper Bound for the Traveling Salesman Problem in Cubic Graphs

TL;DR

This paper establishes a new upper bound of O(1.2553^n) time for solving the Traveling Salesman Problem specifically in cubic graphs, improving upon previous bounds and correcting earlier analysis errors.

Contribution

It introduces a refined algorithmic upper bound for TSP in cubic graphs and corrects prior analysis mistakes to establish a valid complexity bound.

Findings

New upper bound of O(1.2553^n) for TSP in cubic graphs

Modified Eppstein's algorithm achieves the bound

Previous bound of O(1.251^n) was invalid due to analysis errors

Abstract

We provide a new upper bound for traveling salesman problem (TSP) in cubic graphs, i.e. graphs with maximum vertex degree three, and prove that the problem for an $n$-vertex graph can be solved in $O(1.2553^n)$ time and in linear space. We show that the exact TSP algorithm of Eppstein, with some minor modifications, yields the stated result. The previous best known upper bound $O(1.251^n)$ was claimed by Iwama and Nakashima [Proc. COCOON 2007]. Unfortunately, their analysis contains several mistakes that render the proof for the upper bound invalid.