Beating the integrality ratio for s-t-tours in graphs

TL;DR

This paper presents a polynomial-time algorithm for the s-t-path graph TSP with an approximation ratio of 1.497, improving upon the known integrality ratio of 3/2 and introducing new techniques like a novel ear-decomposition and matroid union connection.

Contribution

It introduces the first algorithm to beat the 3/2 integrality ratio for s-t-path graph TSP, using innovative methods such as a new ear-decomposition and a reduction to small-distance instances.

Findings

Achieved an approximation ratio of 1.497 for s-t-path graph TSP.

Developed a new ear-decomposition technique and matroid union connection.

Reduced general instances to small-distance cases for improved approximation.

Abstract

Among various variants of the traveling salesman problem, the s-t-path graph TSP has the special feature that we know the exact integrality ratio, 3/2, and an approximation algorithm matching this ratio. In this paper, we go below this threshold: we devise a polynomial-time algorithm for the s-t-path graph TSP with approximation ratio 1.497. Our algorithm can be viewed as a refinement of the 3/2-approximation algorithm by Seb\H{o} and Vygen [2014], but we introduce several completely new techniques. These include a new type of ear-decomposition, an enhanced ear induction that reveals a novel connection to matroid union, a stronger lower bound, and a reduction of general instances to instances in which s and t have small distance (which works for general metrics).