@online{Antoniadis_arXiv2008.12075,
TITLE = {On the Approximability of the Traveling Salesman Problem with Line Neighborhoods},
AUTHOR = {Antoniadis, Antonios and Kisfaludi-Bak, S{\'a}ndor and Laekhanukit, Bundit and Vaz, Daniel},
LANGUAGE = {eng},
URL = {https://arxiv.org/abs/2008.12075},
EPRINT = {2008.12075},
EPRINTTYPE = {arXiv},
YEAR = {2020},
ABSTRACT = {We study the variant of the Euclidean Traveling Salesman problem where<br>instead of a set of points, we are given a set of lines as input, and the goal<br>is to find the shortest tour that visits each line. The best known upper and<br>lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are<br>$\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is<br>based on a reduction to the group Steiner tree problem.<br> We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$.<br>More generally, this implies that TSP with $k$-dimensional flats does not admit<br>a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a<br>complete classification of the approximability of these problems, as there are<br>known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to<br>give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP<br>with lines does not admit a $(2-\epsilon)$-approximation in $d$ dimensions<br>under the unique games conjecture. On the positive side, we leverage recent<br>results on restricted variants of the group Steiner tree problem in order to<br>give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a<br>running time of $n^{O(\log\log n)}$.<br>},
}