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License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2022.10
URN: urn:nbn:de:0030-drops-161706
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16170/
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Antoniadis, Antonios ; Kisfaludi-Bak, Sándor ; Laekhanukit, Bundit ; Vaz, Daniel

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

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LIPIcs-SWAT-2022-10.pdf (0.9 MB)

Abstract

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in ℝ^d, with d ≥ 3, are NP-hardness and an O(log³ n)-approximation algorithm which is based on a reduction to the group Steiner tree problem.
We show that TSP with lines in ℝ^d is APX-hard for any d ≥ 3. More generally, this implies that TSP with k-dimensional flats does not admit a PTAS for any 1 ≤ k ≤ d-2 unless P = NP, which gives a complete classification regarding the existence of polynomial time approximation schemes for these problems, as there are known PTASes for k = 0 (i.e., points) and k = d-1 (hyperplanes). We are able to give a stronger inapproximability factor for d = O(log n) by showing that TSP with lines does not admit a (2-ε)-approximation in d dimensions under the Unique Games Conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an O(log² n)-approximation algorithm for the problem, albeit with a running time of n^{O(log log n)}.

BibTeX - Entry

@InProceedings{antoniadis_et_al:LIPIcs.SWAT.2022.10,
  author =	{Antoniadis, Antonios and Kisfaludi-Bak, S\'{a}ndor and Laekhanukit, Bundit and Vaz, Daniel},
  title =	{{On the Approximability of the Traveling Salesman Problem with Line Neighborhoods}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{10:1--10:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16170},
  URN =		{urn:nbn:de:0030-drops-161706},
  doi =		{10.4230/LIPIcs.SWAT.2022.10},
  annote =	{Keywords: Traveling Salesman with neighborhoods, Group Steiner Tree, Geometric approximation algorithms}
}

Keywords: Traveling Salesman with neighborhoods, Group Steiner Tree, Geometric approximation algorithms
Collection: 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)
Issue Date: 2022
Date of publication: 22.06.2022

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