An Exact Algorithm for TSP in Degree-3 Graphs via Circuit Procedure and Amortization on Connectivity Structure

TL;DR

This paper introduces an exact, polynomial-space algorithm for solving the Traveling Salesman Problem in degree-3 graphs, achieving a new best exponential time bound through a novel circuit-based branch-and-search method.

Contribution

It presents a simple branch-and-search algorithm with an amortization scheme that improves the time complexity for TSP in degree-3 graphs.

Findings

Achieves an O^*(1.2312^n) time complexity, improving previous bounds.

Uses a measure-and-conquer analysis with amortization over circuit structures.

Employs a branch rule based on cut-circuit structures for efficient search.

Abstract

The paper presents an O^*(1.2312^n)-time and polynomial-space algorithm for the traveling salesman problem in an n-vertex graph with maximum degree 3. This improves the previous time bounds of O^*(1.251^n) by Iwama and Nakashima and O^*(1.260^n) by Eppstein. Our algorithm is a simple branch-and-search algorithm. The only branch rule is designed on a cut-circuit structure of a graph induced by unprocessed edges. To improve a time bound by a simple analysis on measure and conquer, we introduce an amortization scheme over the cut-circuit structure by defining the measure of an instance to be the sum of not only weights of vertices but also weights of connected components of the induced graph.