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.


Related in: MedlinePlus

Illustration of the definition of zw and γw before and after applying the gadget transformations. Thick edges are contributing to zw, dashed edges to γw
© Copyright Policy
Related In: Results  -  Collection


getmorefigures.php?uid=PMC4538985&req=5

Fig5: Illustration of the definition of zw and γw before and after applying the gadget transformations. Thick edges are contributing to zw, dashed edges to γw

Mentions: Let us now proceed to prove Lemma 6. For any non-root in-vertex w let . Basically, if v is the parent of w in T, then zw is the total value of x∗ over all edges connecting v with vertices in the subtree Tw of T determined by w. By equality (1) we have 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_v = \max(0,z_v-2). $$\end{document} Also, let γw be the total of x∗ over all edges connecting vertices in Tw with vertices above v. Note that max(0,1−γv) is essentially by how much f′ falls short of reaching the lower-bound of 1 on arc (v,w). The definitions of zw and γw are illustrated in Fig. 5. Fig. 5


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

Mucha M - Theory Comput Syst (2014)

Illustration of the definition of zw and γw before and after applying the gadget transformations. Thick edges are contributing to zw, dashed edges to γw
© Copyright Policy
Related In: Results  -  Collection

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

Fig5: Illustration of the definition of zw and γw before and after applying the gadget transformations. Thick edges are contributing to zw, dashed edges to γw
Mentions: Let us now proceed to prove Lemma 6. For any non-root in-vertex w let . Basically, if v is the parent of w in T, then zw is the total value of x∗ over all edges connecting v with vertices in the subtree Tw of T determined by w. By equality (1) we have 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_v = \max(0,z_v-2). $$\end{document} Also, let γw be the total of x∗ over all edges connecting vertices in Tw with vertices above v. Note that max(0,1−γv) is essentially by how much f′ falls short of reaching the lower-bound of 1 on arc (v,w). The definitions of zw and γw are illustrated in Fig. 5. Fig. 5

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.


Related in: MedlinePlus