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

Abstract

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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 𝒫 = 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 Δ β -TSP for an appropriate β. In case of the Delta-TSP, we obtain a lower bound of 3813 3812 - ε on the polynomial-time approximability (for any small ε > 0 ), compared to the previous bound of 5381 5380 - ε in [11]. In case of the Δ β -TSP, for the relaxed case ( β > 1 ) we present a lower bound of 3803 + 10 β 3804 + 8 β - ε , and for the sharpened triangle inequality ( 1 2 < β < 1 ), the lower bound is 7611 + 10 β 2 + 5 β 7612 + 8 β 2 + 4 β - ε . The latter result is of interest especially since it shows that the TSP is 𝒜𝒫𝒳 -hard even if one comes arbitrarily close to the trivial case that all edges have the same cost.

How to cite

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Böckenhauer, 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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