  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 $1\le k \leq d-2$ 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 $n^{O(\log\log n)}$.
