On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

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 ${\mathbb{R}}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O({\log }^{3}n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in ${\mathbb{R}}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k\le d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of 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 $(2-\epsilon )$-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 $O({\log }^{2}n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log \log n)}$.

References in corpus (1)

• Approximation algorithms for TSP with neighborhoods in the plane