On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

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(log^3 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 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-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(log^2 n)-approximation algorithm for the problem, albeit with a running time of n^O(loglog n).