Limits...
[Formula: see text]-Approximation for Graphic TSP.

Mucha M - Theory Comput Syst (2014)

Bottom Line: 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 [Formula: see text], even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only [Formula: see text].In this paper, we provide an improved analysis of the approach presented in Mömke and Svensson (FOCS, 560-569, 2011) yielding a bound of [Formula: see text] on the approximation factor, as well as a bound of [Formula: see text] for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.

View Article: PubMed Central - PubMed

Affiliation: Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland.

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 [Formula: see text], even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only [Formula: see text]. 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 [Formula: see text] on the approximation factor, as well as a bound of [Formula: see text] for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.

No MeSH data available.


for the graph above (so-called prism graph) is 3n—simply put xe=1 for all horizontal edges e, and  for the remaining edges. On the other hand, visiting all vertices requires going through 4n−2 edges, and so in this case the integrality gap of LP(G) can approach  with n→∞
© Copyright Policy
Related In: Results  -  Collection


getmorefigures.php?uid=PMC4538985&req=5

Fig1: for the graph above (so-called prism graph) is 3n—simply put xe=1 for all horizontal edges e, and for the remaining edges. On the other hand, visiting all vertices requires going through 4n−2 edges, and so in this case the integrality gap of LP(G) can approach with n→∞

Mentions: One of the natural directions of attacking these problems is to consider special cases and several attempts of this nature has been made. Among the most interesting is 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 G=(V,E) and we need to find a shortest tour that visits each vertex at least once. Yet another equivalent formulation asks for a minimum size Eulerian multigraph spanning V and only using edges of G. Similar equivalent formulations apply to the graphic TSPP case. The reason why these special cases are interesting is that they seem to include the difficult inputs of TSP/TSPP. Not only are they APX-hard (see [5]), but also the standard examples showing that the Held-Karp LP relaxation has a gap of at least in the TSP case and in the TSPP case, are in fact graphic metrics (see Figs. 1 and 2). Fig. 1


[Formula: see text]-Approximation for Graphic TSP.

Mucha M - Theory Comput Syst (2014)

for the graph above (so-called prism graph) is 3n—simply put xe=1 for all horizontal edges e, and  for the remaining edges. On the other hand, visiting all vertices requires going through 4n−2 edges, and so in this case the integrality gap of LP(G) can approach  with n→∞
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4538985&req=5

Fig1: for the graph above (so-called prism graph) is 3n—simply put xe=1 for all horizontal edges e, and for the remaining edges. On the other hand, visiting all vertices requires going through 4n−2 edges, and so in this case the integrality gap of LP(G) can approach with n→∞
Mentions: One of the natural directions of attacking these problems is to consider special cases and several attempts of this nature has been made. Among the most interesting is 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 G=(V,E) and we need to find a shortest tour that visits each vertex at least once. Yet another equivalent formulation asks for a minimum size Eulerian multigraph spanning V and only using edges of G. Similar equivalent formulations apply to the graphic TSPP case. The reason why these special cases are interesting is that they seem to include the difficult inputs of TSP/TSPP. Not only are they APX-hard (see [5]), but also the standard examples showing that the Held-Karp LP relaxation has a gap of at least in the TSP case and in the TSPP case, are in fact graphic metrics (see Figs. 1 and 2). Fig. 1

Bottom Line: 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 [Formula: see text], even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only [Formula: see text].In this paper, we provide an improved analysis of the approach presented in Mömke and Svensson (FOCS, 560-569, 2011) yielding a bound of [Formula: see text] on the approximation factor, as well as a bound of [Formula: see text] for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.

View Article: PubMed Central - PubMed

Affiliation: Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland.

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 [Formula: see text], even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only [Formula: see text]. 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 [Formula: see text] on the approximation factor, as well as a bound of [Formula: see text] for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.

No MeSH data available.