The Thermodynamics of the Travelling Salesman Problem

TL;DR

This paper reviews the application of simulated annealing, a meta-heuristic inspired by thermodynamics, to solve the Traveling Salesman Problem, demonstrating its effectiveness through simulations on different city distributions.

Contribution

It provides a comprehensive review of the thermodynamic formalism of simulated annealing and demonstrates its application to the TSP with simulation results for different scenarios.

Findings

Simulated annealing can find optimal solutions for small TSP instances.

The method yields good approximate solutions for larger, more complex TSP instances.

Simulations show effectiveness in different spatial distributions of cities.

Abstract

In this pedagogical work we reviewed the mathematical formalism and the physical interpretation, based on statistical mechanics, of the meta-heuristics called simulated annealing. Moreover, we presented the mathematical formulation of the algorithm and why it is capable to yield the optimal solution or a good approximated solution of a given problem. Furthermore, we described the travelling salesman problem, showing its interpretation as a Markov Chain and how the simulated annealing can be used to optimize it and we did its simulations for two scenarios. Firstly, for 50 cities distributed around a circle and we found the best solution. Finally, we applied the meta-heuristic in a another instance, with 100 nodes random uniformly distributed in a square, and one shows that it allows finding a good solution.