License: Creative Commons Attribution-NoDerivs 3.0 Unported license (CC BY-ND 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2010.2458
URN: urn:nbn:de:0030-drops-24580
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2458/
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van Nijnatten, Fred ; Sitters, René ; Woeginger, Gerhard J. ; Wolff, Alexander ; de Berg, Mark

The Traveling Salesman Problem under Squared Euclidean Distances

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1001.DeBergMark.2458.pdf (0.3 MB)

Abstract

Let $P$ be a set of points in $\Reals^d$, 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.

BibTeX - Entry

@InProceedings{vannijnatten_et_al:LIPIcs:2010:2458,
  author =	{Fred van Nijnatten and Ren{\'e} Sitters and Gerhard J. Woeginger and Alexander Wolff and Mark de Berg},
  title =	{{The Traveling Salesman Problem under Squared Euclidean Distances}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{239--250},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Jean-Yves Marion and Thomas Schwentick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2458},
  URN =		{urn:nbn:de:0030-drops-24580},
  doi =		{10.4230/LIPIcs.STACS.2010.2458},
  annote =	{Keywords: Geometric traveling salesman problem, power-assignment in wireless networks, distance-power gradient, NP-hard, APX-hard}
}

Keywords: Geometric traveling salesman problem, power-assignment in wireless networks, distance-power gradient, NP-hard, APX-hard
Collection: 27th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2010
Date of publication: 09.03.2010

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