On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

TL;DR

This paper investigates the complexity and approximability of the Traveling Salesman Problem with line neighborhoods in Euclidean space, establishing hardness results and providing an approximation algorithm with specific bounds.

Contribution

It proves APX-hardness for TSP with lines in dimensions three and higher, classifies the approximability for various flat dimensions, and offers an $O( ext{log}^2 n)$-approximation algorithm.

Findings

TSP with lines in $ extbf{R}^d$ is APX-hard for $d extgreater 2$.

No $(2- ext{epsilon})$-approximation exists under the Unique Games Conjecture for certain dimensions.

An $O( ext{log}^2 n)$-approximation algorithm with quasi-polynomial runtime is proposed.

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