The traveling salesman problem for lines and rays in the plane

TL;DR

This paper introduces improved linear-time approximation algorithms for the Euclidean TSP with neighborhoods specifically for line and ray regions, along with a new bound on rectangle perimeter for open curves.

Contribution

It provides the first efficient approximation algorithms for TSPN with line and ray neighborhoods and establishes a tight perimeter bound for open curves.

Findings

Improved approximation ratios for TSPN with lines and rays

Linear-time algorithms developed for these specific neighborhood types

A tight bound on the minimum perimeter of rectangles enclosing open curves

Abstract

In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of $n$ regions (neighborhoods) and we seek a shortest tour that visits each region. In the path variant, we seek a shortest path that visits each region. We present several linear-time approximation algorithms with improved ratios for these problems for two cases of neighborhoods that are (infinite) lines, and respectively, (half-infinite) rays. Along the way we derive a tight bound on the minimum perimeter of a rectangle enclosing an open curve of length $L$.