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27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Nancy : France (2010)

The Traveling Salesman Problem Under Squared Euclidean Distances

Mark De Berg (  Auteur correspondant ) 1, Fred Van Nijnatten 1, Rene Sitters 2, Gerhard J. Woeginger 1, Alexander Wolff 3

(2010)

Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by TSP($d,\alpha$). We design a 5-approximation algorithm for TSP(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\alpha}/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-TSP of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-TSP$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-TSP$(d, \alpha)$ is APX-hard if $d\ge3$ and $\alpha>1$. The APX-hardness proof carries over to TSP$(d, \alpha)$ for the same parameter ranges.

1 :	Department of Mathematics and Computer Science
	Technische Universiteit Eindhoven
2 :	Faculty of Economics and Business Administration,
	Vrije Universiteit Amsterdam
3 :	Lehrstuhl für Informatik I
	Universität Würzburg

Domaine

:

Informatique/Géométrie algorithmique Informatique/Complexité

Mots Clés : Geometric traveling salesman problem – power-assignment in wireless networks – distance-power gradient – NP-hard – APX-hard

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Soumis le : Mardi 9 Février 2010, 16:48:49

Dernière modification le : Mardi 9 Février 2010, 16:50:48