On the Nearest Neighbor Algorithm for Mean Field Traveling Salesman Problem

TL;DR

This paper analyzes the asymptotic behavior of the nearest neighbor tour in the mean field traveling salesman problem with i.i.d. intercity distances, showing it scales as log n under certain conditions.

Contribution

It extends known results for Euclidean TSP to the mean field case, deriving the limiting behavior based on the distribution's density at zero.

Findings

Nearest neighbor tour length scales as log n for exponential distributions.

Asymptotic behavior depends on the density of the distribution at zero.

Results generalize to broader distribution classes based on their scaling properties.

Abstract

In this work we consider the mean field traveling salesman problem, where the intercity distances are taken to be i.i.d. with some distribution $F$. This paper focus on the \emph{nearest neighbor tour} which is to move to the nearest non-visited city and we show that under some conditions on $F$, which are satisfied by exponential distribution with constant mean, the total length of the nearest neighbor tour, asymptotically almost surely scales as $\log n$. Similar result is known for Euclidean TSP and nearest neighbor tour. We further derive the limiting behavior of the total length of the nearest neighbor tour for more general distribution function $F$ and show that its asymptotic properties are determined by the scaling properties of the density of $F$ at 0.