A Neural Network for the Travelling Salesman Problem with a Well Behaved Energy Function

We present and analyze a Self Organizing Feature Map (SOFM) for the NP-complete problem of the travelling salesman (TSP): finding the shortest closed path joining N cities. Since the SOFM has discrete input patterns (the cities of the TSP) one can examine its dynamics analytically. We show that, with a particular choice of the distance function for the net, the energy associated to the SOFM has its absolute minimum at the shortest TSP path. Numerical simulations confirm that this distance augments performances. It is curious that the distance function having this property combines the distances of the neuron and of the weight spaces.