The stochastic traveling salesman problem: Finite size scaling and the cavity prediction

TL;DR

This paper investigates the stochastic traveling salesman problem using the cavity method, validating its predictions through simulations and revealing insights into the distribution of links in optimal tours.

Contribution

It applies the cavity method to the stochastic TSP, providing numerical validation of the replica symmetric solution and exploring the distribution of links in optimal tours.

Findings

Excellent agreement between cavity predictions and simulations.

Numerical evidence supports the replica symmetric solution.

Surprising distribution pattern of kth-nearest neighbor links.

Abstract

We study the random link traveling salesman problem, where lengths l_ij between city i and city j are taken to be independent, identically distributed random variables. We discuss a theoretical approach, the cavity method, that has been proposed for finding the optimal tour length over this random ensemble, given the assumption of replica symmetry. Using finite size scaling and a renormalized model, we test the cavity predictions against the results of simulations, and find excellent agreement over a range of distributions. We thus provide numerical evidence that the replica symmetric solution to this problem is the correct one. Finally, we note a surprising result concerning the distribution of kth-nearest neighbor links in optimal tours, and invite a theoretical understanding of this phenomenon.