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oai:drops-oai.dagstuhl.de:16170 2024-03-06T10:57:04Z ddc:004 open_access

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods Antoniadis, Antonios Kisfaludi-Bak, Sándor Laekhanukit, Bundit Vaz, Daniel Traveling Salesman with neighborhoods Group Steiner Tree Geometric approximation algorithms 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)}. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Antonios Antoniadis and Sándor Kisfaludi-Bak and Bundit Laekhanukit and Daniel Vaz 2022 Is Part Of LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SWAT.2022.10 urn:nbn:de:0030-drops-161706 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.10 eng https://creativecommons.org/licenses/by/4.0/legalcode