# \frac{13}{9}-Approximation for Graphic TSP

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Theory of Computing Systems

, Volume 55, Issue 4, pp 640–657

First Online: 07 December 2012

Abstract

The Travelling Salesman Problem is one of the fundamental and intensively studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides’s algorithm with an approximation factor of \\frac{3}{2}\, even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only \\frac{4}{3}\. Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis Gharan et al. FOCS, 550–559, 2011, and then by Mömke and Svensson FOCS, 560–569, 2011. In this paper, we provide an improved analysis of the approach presented in Mömke and Svensson FOCS, 560–569, 2011 yielding a bound of \\frac{13}{9}\ on the approximation factor, as well as a bound of \\frac{19}{12}+\varepsilon\ for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.

KeywordsApproximation algorithms Travelling salesman problem This work was partially supported by the Polish Ministry of Science grant N206 355636 and by the ERC StG project PAAl no. 259515. Preliminary version of this paper was announced at STACS 2012 11.

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Author: **Marcin Mucha**

Source: https://link.springer.com/