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.
@InProceedings{vannijnatten_et_al:LIPIcs.STACS.2010.2458, author = {van Nijnatten, Fred and Sitters, Ren\'{e} and Woeginger, Gerhard J. and Wolff, Alexander and de Berg, Mark}, 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 = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.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} }
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