13/9 -Approximation for Graphic TSP

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 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. (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 13/9 on the approximation factor, as well as a bound of 19/12+ε for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.