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 , 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.
One of the natural directions of attacking these problem is to consider special cases and several attempts of this nature has been made. The most interesting one is by far the graphic TSP TSPP, where we assume that the given metric is the shortest path metric of an undirected graph. Equivalently, in graphic TSP we are given an undirected graph @math and we need to find a shortest tour that visits each vertex . Yet another formulation would ask for a minimum size Eulerian multigraph spanning @math and only using edges of @math . Similar formulations apply to the graphic TSPP case. The reason why these special cases are very interesting is that they seem to include the difficult inputs of TSP TSPP. Not only are they APX-hard (see @cite_7 ), but also the standard examples showing that the Held-Karp relaxation has a gap of at least @math in the TSP case and @math in the TSPP case, are in fact graphic metrics.
@cite_7
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We consider the special case of the traveling salesman problem (TSP) in which the distance metric is the shortest-path metric of a planar unweighted graph. We present a polynomial-time approximation scheme (PTAS) for this problem.