Replica Symmetry and Replica Symmetry Breaking for the Traveling Salesperson Problem

TL;DR

This paper investigates the energy landscape of the Traveling Salesperson Problem using exact and excited states, finding no evidence of replica symmetry breaking in Euclidean TSP but potential signs in the (1,2)-TSP.

Contribution

It introduces a novel linear programming approach to study the energy landscape of TSP and compares different ensembles, providing new insights into replica symmetry properties.

Findings

Euclidean TSP shows no signs of replica symmetry breaking.

(1,2)-TSP exhibits signatures that may indicate broken replica symmetry.

The approach enables detailed analysis of excited states in TSP.

Abstract

We study the energy landscape of the Traveling Salesperson problem (TSP) using exact ground states and a novel linear programming approach to generate excited states with closely defined properties. We look at four different ensembles, notably the classic finite dimensional Euclidean TSP and the mean-field-like (1,2)-TSP, which has its origin directly in the mapping of the Hamiltonian circuit problem on the TSP. Our data supports previous conjectures that the Euclidean TSP does not show signatures of replica symmetry breaking neither in two nor in higher dimension. On the other hand the (1,2)-TSP exhibits some signature which does not exclude broken replica symmetry, making it a candidate for further studies in the future.