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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.


Gadget replacement for non-leaf vertices of G other than the root
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Fig3: Gadget replacement for non-leaf vertices of G other than the root

Mentions: Let v be any non-root vertex of G having l children: w1,…,wl in T. We introduce l new vertices v1,…,vl and replace the tree-arc (v,wj) by tree-arcs (v,vj) and (vj,wj) for j=1,…,l. We also redirect to vj all the back-arcs leaving the subtree rooted at wj and entering v (see Fig. 3). We will call the new vertices and the root in-vertices and the remaining vertices out-vertices. We will also denote the set of all in-vertices by , and the set of in-vertices in the gadget corresponding to v by . Notice that all the back-arcs go from out-vertices to in-vertices, and that each in-vertex has exactly one outgoing arc (for the root vertex this follows from 2-vertex connectivity). Fig. 3


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

Mucha M - Theory Comput Syst (2014)

Gadget replacement for non-leaf vertices of G other than the root
© Copyright Policy
Related In: Results  -  Collection

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

Fig3: Gadget replacement for non-leaf vertices of G other than the root
Mentions: Let v be any non-root vertex of G having l children: w1,…,wl in T. We introduce l new vertices v1,…,vl and replace the tree-arc (v,wj) by tree-arcs (v,vj) and (vj,wj) for j=1,…,l. We also redirect to vj all the back-arcs leaving the subtree rooted at wj and entering v (see Fig. 3). We will call the new vertices and the root in-vertices and the remaining vertices out-vertices. We will also denote the set of all in-vertices by , and the set of in-vertices in the gadget corresponding to v by . Notice that all the back-arcs go from out-vertices to in-vertices, and that each in-vertex has exactly one outgoing arc (for the root vertex this follows from 2-vertex connectivity). Fig. 3

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.