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

Abstract

-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. 1 -Introduction Solving difficult problems is a natural arena for a would-be new calculus paradigm like that of neural networks. One can delineate a sharper image of their potential with respect to the blurred image obtained in simpler problems. Here we tackle the Travelling Salesman Problem (TSP, see [Lawler 1985], [Johnson 1990]) with a Self Organizing Feature Map (SOFM). This approach, proposed by [Angéniol 1988] and [Favata 1991], started to produce respectable performances with the elimination of the non-injective outputs produced by the SOFM [Budinich 1995]. In this paper we further improve its performances by choosing a suitable distance function for the SOFM. An interesting feature is that this net is open to analytical inspection down to a level that is not usually reachable [Ritter 1992]. This happens because the input patterns of the SOFM, namely the cities of the TSP, are discrete. As a consequence we can show that the energy function, associated with SOFM learning, has its absolute minimum in correspondence to the shortest TSP path. In what follows we start with a brief presentation of the working principles of this net and of its basic theoretical analysis (section 2). In section 3 we propose a new distance function for the network and show its theoretical advantages while section 4 contains numerical results. The appendix contains the detailed description of parameters needed to reproduce these results.