The Maximum Scatter TSP on a Regular Grid

TL;DR

This paper introduces an efficient algorithm for the maximum scatter TSP on a grid, optimizing the shortest edge in the tour, with applications in manufacturing, and provides optimal or near-optimal solutions for grid instances.

Contribution

It extends an existing line-based algorithm to 2D grids, offering a linear-time optimal solution in some cases and a strong approximation for the worst case.

Findings

The $ extsc{Weave}(m,n)$ algorithm computes optimal tours in linear time for certain grid sizes.

It is asymptotically optimal for the maximum scatter TSP on grids.

Provides a $rac{ ext{sqrt}(10)}{5}$-approximation for the 3x4 grid, the worst case.

Abstract

In the maximum scatter traveling salesman problem the objective is to find a tour that maximizes the shortest distance between any two consecutive nodes. This model can be applied to manufacturing processes, particularly laser melting processes. We extend an algorithm by Arkin et al. that yields optimal solutions for nodes on a line to a regular $m \times n$-grid. The new algorithm $\textsc{Weave}(m,n)$ takes linear time to compute an optimal tour in some cases. It is asymptotically optimal and a $\frac{\sqrt{10}}{5}$-approximation for the $3\times 4$-grid, which is the worst case.