The random link approximation for the Euclidean traveling salesman problem

TL;DR

This paper analyzes the Euclidean TSP with randomly distributed cities, deriving average tour lengths and validating the random link approximation's accuracy across dimensions, providing analytical predictions and conjectures.

Contribution

It introduces the random link approximation for the Euclidean TSP and validates its accuracy, offering analytical predictions for average tour lengths in various dimensions.

Findings

Random link approximation closely matches Euclidean TSP results in low dimensions.

Derived analytical predictions for average tour lengths using cavity equations.

Conjectured asymptotic behavior of the TSP length in high dimensions.

Abstract

The traveling salesman problem (TSP) consists of finding the length of the shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where the cities are distributed randomly and independently in a d-dimensional unit hypercube. Working with periodic boundary conditions and inspired by a remarkable universality in the kth nearest neighbor distribution, we find for the average optimum tour length <L_E> = beta_E(d) N^{1-1/d} [1+O(1/N)] with beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive analytical predictions for these quantities using the random link approximation, where the lengths between cities are taken as independent random variables. From the ``cavity'' equations developed by Krauth, Mezard and Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3, numerical results show that the random link approximation is a good one, with a discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we argue that the approximation is exact up to O(1/d^2) and give a conjecture for beta_E(d), in terms of a power series in 1/d, specifying both leading and subleading coefficients.