The random link approximation for theEuclidean

The traveling salesman problem (TSP) consists of nding 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 nd for the average optimum tour length hL E i = 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, M ezard and Parisi, we calculate the associated random link values 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 E (d) and RL (d). For large d, we argue that the approximation is exact up to O(1=d 2) and give a conjecture for E (d), in terms of a power series in 1=d, specifying both leading and subleading coeecients.