13/9-approximation for Graphic TSP

TL;DR

This paper improves the approximation ratio for the Graphic TSP, achieving a 13/9 bound by refining previous analysis, and extends results to the Traveling Salesman Path Problem in graphic metrics.

Contribution

It provides a tighter analysis of existing algorithms for Graphic TSP, reducing the approximation factor from previous bounds, and extends the approach to related path problems.

Findings

Achieved a 13/9 approximation bound for Graphic TSP.

Extended analysis to the Traveling Salesman Path Problem with a 19/12+epsilon bound.

Improved understanding of approximation limits for graphic metrics.

Abstract

The Travelling Salesman Problem is one the most fundamental and most studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides's algorithm with approximation factor of 3/2, even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only 4/3. Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis Gharan et al., and then by Momke and Svensson. In this paper, we provide an improved analysis for the approach introduced by Momke and Svensson yielding a bound of 13/9 on the approximation factor, as well as a bound of 19/12+epsilon for any epsilon>0 for a more general Travelling Salesman Path Problem in graphic metrics.