Approximation hardness of graphic TSP on cubic graphs

Karpiński, Marek; Schmied, Richard

doi:10.1051/ro/2014062

Cited by 7 publications

(12 citation statements)

References 15 publications

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“…Our overall approach is to construct appropriate reductions from the (1, 2)-Metric TSP to (1, 2)-metric instances whose underlying graph is a PLG. We make use of the lower bound constructions developed in [KS13a] (see also [EK06;Lam12;KS13b;KLS13]). The reduction in [KS13a] starts from a special bounded occurrence optimization problem.…”

Section: Lower Boundsmentioning

confidence: 99%

“…The above result is used in the following reduction from Hybrid to Subcubic (1, 2)-TSP due to Karpinski and Schmied [KS13a]. Later we present our new reduction from slightly modified Subcubic (1, 2)-TSP instances to (1, 2)-TSP instances with underlying PLG.…”

Section: Lower Boundsmentioning

confidence: 99%

“…Reduction from Hybrid to Subcubic (1, 2)-TSP The reduction starts from Hybrid instances with 60v 2-equations, 2v 3-equations and 42v variables. In [KS13a], these instances have been mapped to (1, 2)-TSP instances G 12 SC with 60v · 8 + 2v · (9 · 8 + 6 · 3 + 2 · 3) = v · 672 nodes. Our aim is to modify these instances slightly in order to fit them into a power law degree distribution.…”

Section: Lower Boundsmentioning

confidence: 99%

“…Let us describe the simulating gadget for a 3-equation x ⊕ y ⊕ z = 0, as constructed in [KS13a]. Given an equation of the form x ⊕ y ⊕ z = 0, three clauses (x ∨ a 1 ∨ a 2 ), (y ∨ a 2 ∨ a 3 ) and (z ∨ a 1 ∨ a 3 ) are created.…”

Section: Lower Boundsmentioning

confidence: 99%

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Approximability of the vertex cover problem in power-law graphs

Gast

Hauptmann

2014

Theoretical Computer Science

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In this paper we study the special case of Graphic TSP where the underlying graph is a power law graph (PLG). We give a refined analysis of some of the current best approximation algorithms and show that an improved approximation ratio can be achieved for certain ranges of the power law exponent β. For the value of power law exponent β = 1.5 we obtain an approximation ratio of 1.34 for Graphic TSP. Moreover we study the (1, 2)-TSP with the underlying graph of 1-edges being a PLG. We show improved approximation ratios in the case of underlying deterministic PLGs for β greater than 1.666. For underlying random PLGs we further improve the analysis and show even better expected approximation ratio for the range of β between 1 and 3.5. On the other hand we prove the first explicit inapproximability bounds for (1, 2)-TSP for an underlying power law graph., where E k is the expected lower bound on the number of 2-edges in an optimum tour.On the other hand we show that for β > 1, the Power Law (1, 2)-TSP is NP-hard to approximate within approximation ratio (ζ(β)+ 1 /2)·3 β−1 ·2·(β−1)·354+1 (ζ(β)+ 1 /2)·3 β−1 ·2·(β−1)·354 . This gives an approximation lower bound of 1.00086 for β = 1.2 and of 1.0012 for β = 1.1. Organization of the PaperIn Section 2 we give an outline of the methods and constructions used in the paper. Section 3 provides the definition of the PLG model due to [ACL01] and related notations. In Section 4 we describe our results for the Graphic TSP on power law graphs. In Section 5 we present our results on the (1, 2)-TSP with underlying power law graph. Section 5.1 contains the analysis of the algorithm from [PY93] for deterministic power law graphs. Section 5.2 deals with the case of a large power law exponent. In Section 5.3 we consider the case when the underlying graph is a random PLG. Approximation lower bounds are given in Section 5.4. 2 8

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Section: Lower Boundsmentioning

confidence: 99%

Section: Lower Boundsmentioning

confidence: 99%

Section: Lower Boundsmentioning

confidence: 99%

Section: Lower Boundsmentioning

confidence: 99%

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Approximability of the vertex cover problem in power-law graphs

Gast

Hauptmann

2014

Theoretical Computer Science

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“…For the case B = 8, Engebretsen and Karpinski [12] gave an inapproximability bound of 135/134. For corresponding approximation bounds for Graphic TSP see [17].…”

Section: Introductionmentioning

confidence: 99%