Probabilistic Analysis of Edge Elimination for Euclidean TSP

TL;DR

This paper analyzes the effectiveness of two edge elimination methods for Euclidean TSP on random points, showing that one reduces edges to linear in number while the other leaves quadratic edges, impacting TSP preprocessing efficiency.

Contribution

It provides a probabilistic analysis of two edge elimination techniques for Euclidean TSP on random point sets, quantifying their expected edge reductions.

Findings

Hougardy and Schroeder's method reduces edges to Θ(n)

Jonker and Volgenant's method leaves Θ(n^2) edges

Analysis applies to points with bounded density in the unit square

Abstract

One way to speed up the calculation of optimal TSP tours in practice is eliminating edges that are certainly not in the optimal tour as a preprocessing step. In order to do so several edge elimination approaches have been proposed in the past. In this work we investigate two of them in the scenario where the input consists of $n$ independently distributed random points in the 2-dimensional unit square with bounded density function from above and below by arbitrary positive constants. We show that after the edge elimination procedure of Hougardy and Schroeder the expected number of remaining edges is $\Theta(n)$, while after that the non-recursive part of Jonker and Volgenant the expected number of remaining edges is $\Theta(n^2)$.