A PTAS for Travelling Salesman Problem with Neighbourhoods Over Parallel Line Segments of Similar Length

TL;DR

This work delivers a PTAS for Euclidean TSPN when neighbourhoods are parallel line segments of similar length, achieving a (1+ε)-approximation in time n^{O(λ/ε^3)}. Building on Arora’s hierarchical dissection, the authors introduce a dropping technique that removes crossing segments at horizontal dissection lines and prove that the resulting loss is only O(ε·OPT), enabling independent subproblems connected by portals. A key technical contribution is the construction of near-optimal structured solutions with bounded shadow per strip, secured via detailed structural lemmas about zig-zags, sinks, and reflection points, which in turn underpin an outer DP over a quad-tree and an inner DP for base cases. The result settles the approximation landscape for parallel unit-length segments and extends to bounded-length ratios, with implications for axis-parallel variants and related geometric TSPN problems. The proposed approach combines geometric packing, shadow analysis, and carefully designed DPs to achieve a provably close-to-optimal tour in polynomial time for fixed ε and λ.

Abstract

We consider the Travelling Salesman Problem with Neighbourhoods (TSPN) on the Euclidean plane ($\mathbb{R}^2$) and present a Polynomial-Time Approximation Scheme (PTAS) when the neighbourhoods are parallel line segments with lengths between $[1, λ]$ for any constant value $λ\ge 1$. In TSPN (which generalizes classic TSP), each client represents a set (or neighbourhood) of points in a metric and the goal is to find a minimum cost TSP tour that visits at least one point from each client set. In the Euclidean setting, each neighbourhood is a region on the plane. TSPN is significantly more difficult than classic TSP even in the Euclidean setting, as it captures group TSP. A notable case of TSPN is when each neighbourhood is a line segment. Although there are PTASs for when neighbourhoods are fat objects (with limited overlap), TSPN over line segments is APX-hard even if all the line segments have unit length. For parallel (unit) line segments, the best approximation factor is $3\sqrt2$ from more than two decades ago [DM03]. The PTAS we present in this paper settles the approximability of this case of the problem. Our algorithm finds a $(1 + ε)$-factor approximation for an instance of the problem for $n$ segments with lengths in $ [1,λ] $ in time $ n^{O(λ/ε^3)} $.

Figures (27)

• Figure 1: A potential arrangement of line segments where the solution has a large shadow
• Figure 2: An optimum solution to these special case instances intersects with any vertical line at most twice
• Figure 3: An example of an instance of the problem with bounding box of height at most 5. We have at most 3 strips in this case and maximum shadow of the given solution is 6
• Figure 4: If $p_j$ isn't a tip of $s_i$, then $\ell_j, \ell_{j + 1}$ must be collinear
• Figure 5: In range $I$, $P_1$ is above $P_2, P_3$, and $P_2$ is above $P_3$

Theorems & Definitions (89)

• Theorem 1
• Theorem 2
• Definition 1
• Theorem 3
• Definition 2
• Lemma 1
• proof
• Definition 3
• Definition 4
• Lemma 2