On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

TL;DR

This work analyzes the Traveling Salesman Problem with Neighborhoods (TSPN) where neighborhoods are lines in $\mathbb{R}^d$. It establishes APX-hardness for line neighborhoods in dimensions $d\ge 3$, including a concrete 3D construction that rules out a PTAS, and it shows stronger inapproximability in $d=O(\log n)$ under the Unique Games Conjecture. On the algorithmic side, the paper combines Arora’s Euclidean TSP framework with a modern GST-based approach to obtain an $O(\log^2 n)$-approximation in quasi-polynomial time, via a discrete neighborhood reduction and a STGST formulation. Together, these results provide a near-complete classification of the approximability of TSPN for line neighborhoods and higher-dimensional flats, highlighting a substantial gap between hardness and current quasi-polynomial–time methods and pointing to open questions about constant-factor approximations in 3D.

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 $(2-ε)$-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 $n^{O(\log\log n)}$.

Figures (4)

• Figure 1: Left: Overview of a basic construction with a cube. Right: The optimal tour must visit all points of $Q$, and it makes detours to some points $p_v$.
• Figure 2: (i) Cross-section given by a plane perpendicular to $\bar{e^a}$. (The segment $\bar{e^a}$ appears as a point, and the plane $H_a$ as a line in this picture.) All lines of $\bar{\mathcal{L}}$ intersect such a plane in the gray area. (ii) Defining $Q^a$ within the plane $H_a$, so that all points have distance at least $\delta^*$ from $\bar{e^a}$. (iii) Defining $Q$ so that it has all the required properties. The cylinder $\mathcal{Y}$ is perpendicular to the plane $x+y+z=0$, a plane to which all points of the construction are close to. The "triangle" defined by the skew lines $\bar{e}^a$ wraps around the cylinder $\mathcal{Y}$.
• Figure 3: Bounding the distance of the skew lines $\ell,\ell'\in \mathcal{L}$.
• Figure 4: Defining the lines of a point gadget for a point $q$ using a grid $\Gamma$.

Theorems & Definitions (22)

• Theorem 1
• Corollary 2
• Theorem 3
• Corollary 4
• Theorem 5
• Lemma 6
• Corollary 7
• Claim 8
• Claim 9
• Lemma 10: Point gadget