Mean field matching and TSP in pseudo-dimension 1

TL;DR

This paper rigorously confirms the limit length of a traveling salesman tour in a complete graph with uniform edges, using a unique solution to an integral equation derived from the cavity method, bridging physics conjectures and mathematical proof.

Contribution

It proves the uniqueness of the integral equation's solution and provides a rigorous derivation of the TSP limit in the pseudo-dimension 1 model.

Findings

Confirmed the limit length of TSP tour as 2.0415.

Established the uniqueness of the integral equation's solution.

Provided a rigorous derivation of the TSP limit.

Abstract

Recent work on optimization problems in random link models has verified several conjectures originating in statistical physics and the replica and cavity methods. In particular the numerical value 2.0415 for the limit length of a traveling salesman tour in a complete graph with uniform $[0,1]$ edge lengths has been established. In this paper we show that the crucial integral equation obtained with the cavity method has a unique solution, and that the limit ground state energy obtained from this solution agrees with the rigorously derived value. Moreover, the method by which we establish uniqueness of the solution turns out to yield a new completely rigorous derivation of the limit.