# Improved Lower Bounds on the Approximability of the Traveling Salesman Problem

Hans-Joachim Böckenhauer; Sebastian Seibert

RAIRO - Theoretical Informatics and Applications (2010)

- Volume: 34, Issue: 3, page 213-255
- ISSN: 0988-3754

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topBöckenhauer, Hans-Joachim, and Seibert, Sebastian. "Improved Lower Bounds on the Approximability of the Traveling Salesman Problem." RAIRO - Theoretical Informatics and Applications 34.3 (2010): 213-255. <http://eudml.org/doc/222088>.

@article{Böckenhauer2010,

abstract = {
This paper deals with lower bounds on the approximability of different subproblems of the Traveling
Salesman Problem (TSP) which is known not to admit any polynomial time approximation algorithm in general
(unless $\mathcal\{P\}=\mathcal\{NP\}$). First of all, we present an improved lower bound for the Traveling Salesman Problem
with Triangle Inequality, Delta-TSP for short. Moreover our technique, an extension of the method of
Engebretsen [11], also applies to the case of relaxed and sharpened triangle inequality,
respectively, denoted $\Delta_\beta$-TSP for an appropriate β. In case of the
Delta-TSP, we obtain a lower
bound of $\frac\{3813\}\{3812\}-\varepsilon$ on the polynomial-time approximability (for any small
$\varepsilon> 0$), compared to the previous bound of $\frac\{5381\}\{5380\}-\varepsilon$ in [11]. In
case of the $\Delta_\beta$-TSP, for the relaxed case ($\beta> 1$) we present a lower bound of
$\frac\{3803+10\beta\}\{3804+8\beta\}-\varepsilon$, and for the sharpened triangle inequality
($\frac\{1\}\{2\}< \beta< 1$), the lower bound is
$\frac\{7611+10\beta^2+5\beta\}\{7612+8\beta^2+4\beta\}-\varepsilon$. The latter result is of interest
especially since it shows that the TSP is $\mathcal\{APX\}$-hard even if one comes arbitrarily close to the trivial
case that all edges have the same cost.
},

author = {Böckenhauer, Hans-Joachim, Seibert, Sebastian},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Approximation algorithms; Traveling Salesman Problem.; traveling salesman problem},

language = {eng},

month = {3},

number = {3},

pages = {213-255},

publisher = {EDP Sciences},

title = {Improved Lower Bounds on the Approximability of the Traveling Salesman Problem},

url = {http://eudml.org/doc/222088},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Böckenhauer, Hans-Joachim

AU - Seibert, Sebastian

TI - Improved Lower Bounds on the Approximability of the Traveling Salesman Problem

JO - RAIRO - Theoretical Informatics and Applications

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 3

SP - 213

EP - 255

AB -
This paper deals with lower bounds on the approximability of different subproblems of the Traveling
Salesman Problem (TSP) which is known not to admit any polynomial time approximation algorithm in general
(unless $\mathcal{P}=\mathcal{NP}$). First of all, we present an improved lower bound for the Traveling Salesman Problem
with Triangle Inequality, Delta-TSP for short. Moreover our technique, an extension of the method of
Engebretsen [11], also applies to the case of relaxed and sharpened triangle inequality,
respectively, denoted $\Delta_\beta$-TSP for an appropriate β. In case of the
Delta-TSP, we obtain a lower
bound of $\frac{3813}{3812}-\varepsilon$ on the polynomial-time approximability (for any small
$\varepsilon> 0$), compared to the previous bound of $\frac{5381}{5380}-\varepsilon$ in [11]. In
case of the $\Delta_\beta$-TSP, for the relaxed case ($\beta> 1$) we present a lower bound of
$\frac{3803+10\beta}{3804+8\beta}-\varepsilon$, and for the sharpened triangle inequality
($\frac{1}{2}< \beta< 1$), the lower bound is
$\frac{7611+10\beta^2+5\beta}{7612+8\beta^2+4\beta}-\varepsilon$. The latter result is of interest
especially since it shows that the TSP is $\mathcal{APX}$-hard even if one comes arbitrarily close to the trivial
case that all edges have the same cost.

LA - eng

KW - Approximation algorithms; Traveling Salesman Problem.; traveling salesman problem

UR - http://eudml.org/doc/222088

ER -

## References

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## Citations in EuDML Documents

top- Marc Demange, Vangelis Paschos, Autour de nouvelles notions pour l'analyse des algorithmes d'approximation : de la structure de NPO à la structure des instances
- Marc Demange, Vangelis Paschos, Autour de nouvelles notions pour l’analyse des algorithmes d’approximation : de la structure de NPO à la structure des instances

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