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[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 each of the three tight cases, a corresponding part of G and the solution to LP(G) is shown. The bold edges are the tree edges, the remaining edges are back-edges. ε is a very small number
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Fig4: For each of the three tight cases, a corresponding part of G and the solution to LP(G) is shown. The bold edges are the tree edges, the remaining edges are back-edges. ε is a very small number

Mentions: To handle the case of u∗>0 we need another (almost) tight case in the proof of Theorem 10 which occurs when ui is close to ei and ei is relatively large. In this case the value of the expression is clearly close to 0. This corresponds to using items of the form xi=1,li=1 and arbitrary ui. For such elements we have ei=⌈ui+1⌉ and so \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_i + \frac{1}{6}(e_i - u_i) \le u_i + \frac{1}{3}, $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\operatorname{val}_i = l_i + u_i - 1 = u_i, $$\end{document} so the difference between the two is at most . By combining the three types of items described, we can clearly construct C as required for any n and u∗. Figure 4 illustrates the three tight cases directly in terms of the corresponding solutions of LP(G). Fig. 4


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

Mucha M - Theory Comput Syst (2014)

For each of the three tight cases, a corresponding part of G and the solution to LP(G) is shown. The bold edges are the tree edges, the remaining edges are back-edges. ε is a very small number
© Copyright Policy
Related In: Results  -  Collection

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Fig4: For each of the three tight cases, a corresponding part of G and the solution to LP(G) is shown. The bold edges are the tree edges, the remaining edges are back-edges. ε is a very small number
Mentions: To handle the case of u∗>0 we need another (almost) tight case in the proof of Theorem 10 which occurs when ui is close to ei and ei is relatively large. In this case the value of the expression is clearly close to 0. This corresponds to using items of the form xi=1,li=1 and arbitrary ui. For such elements we have ei=⌈ui+1⌉ and so \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_i + \frac{1}{6}(e_i - u_i) \le u_i + \frac{1}{3}, $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\operatorname{val}_i = l_i + u_i - 1 = u_i, $$\end{document} so the difference between the two is at most . By combining the three types of items described, we can clearly construct C as required for any n and u∗. Figure 4 illustrates the three tight cases directly in terms of the corresponding solutions of LP(G). Fig. 4

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.