Formula: see text]-Approximation for Graphic TSP.

Marcin Mucha

doi:10.1007/s00224-012-9439-7

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.

Highlights

Introduction and Related WorkThe Travelling Salesman Problem (TSP) is one the fundamental and intensively studied problems in combinatorial optimization, and approximation algorithms in particular
The goal is to find a path from s to t visiting each point exactly once, except if s and t are the same point in which case it can be visited twice
The circulation used in [8] consists of two parts: the “core” part based on an optimal extreme point solution to the Held-Karp LP relaxation of TSP, and the “correction” part that adds enough flow to the core part to make it feasible

Summary

Introduction and Related Work

The Travelling Salesman Problem (TSP) is one the fundamental and intensively studied problems in combinatorial optimization, and approximation algorithms in particular. Problem (TSPP), in addition to a metric (V , d) we are given two points s, t ∈ V and the goal is to find a path from s to t visiting each point exactly once, except if s and t are the same point in which case it can be visited twice (this is when TSPP reduces to TSP) For this problem, the best approximation algorithm known is that of Hoogeveen [7]. In graphic TSP we are given an undirected graph G = (V , E) and we need to find a shortest tour that visits each vertex at least once Another equivalent formulation asks for a minimum size Eulerian multigraph spanning V and only using edges of G.

Our Results

Organization of the Paper

Preliminaries

Applications to Graphic TSP and TSPP