Neural Nets WIRN VIETRI-98 pp 273-282 | Cite as

# Solving the Travelling Salesman Problem using the Kohonen Network Incorporating Explicit Statistics

## Abstract

In this paper we introduce a new self-organizing neural network, the Kohonen Network Incorporating Explicit Statistics (KNIES) that is based on Kohonen’s Self-Organizing Map (SOM). The primary difference between the SOM and the KNIES is the fact that every iteration in the training phase includes *two* distinct modules — the attracting module and the dispersing module. As a result of the newly introduced dispersing module the neurons maintain the overall statistical properties of the data points. Thus, although in SOM the neurons individually find their places both statistically and topologically, in KNIES they collectively maintain their mean to be the mean of the data points which they represent. The new scheme has been used to solve the Travelling Salesman Problem (TSP). Experimental results for problems taken from TSPLIB [13] indicate that it is a very accurate NN strategy for the TSP — probably the most accurate neural solutions available in the literature.

## Keywords

Travelling Salesman Problem Travel Salesman Problem Winner Neuron Activation Bubble Transient Distribution## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Altınel, İ K., B. J. Oommen and N. Aras, (1997), “Vector Quantization for Arbitrary Distance Function Estimation”,
*INFORMS Journal of Computing*, Vol. 9, 1997, pp. 439–451.MATHCrossRefGoogle Scholar - [2]N. Aras, B. J. Oommen and Altmel, İ K., (1997), “The Kohonen Network Incorporating Explicit Statistics and Its Application to the Traveling Salesman Problem”. Unabridged version of this paper.Google Scholar
- [3]Angéniol, B., C. Vaubois, and J. Y. Le Texier, (1988), “Self-Organizing Feature Maps and the Travelling Salesman Problem,”
*Neural Networks*1, pp. 289–293.CrossRefGoogle Scholar - [4]Burke, L. I. and P. Damany, (1992), “The Guilty Net for the Travelling Salesman Problem,”
*Computers & Operations Research*19, pp. 255–265.MATHCrossRefGoogle Scholar - [5]Duda, R. O. and P. E. Hart, (1973),
*Pattern Classification and Scene Analysis*, Wiley, Chichester.MATHGoogle Scholar - [6]Fort, J. C, (1988), “Solving a Combinatorial Problem via Self-Organizing Process: An Application of the Kohonen Algorithm to the travelling Salesman Problem,
*Biological Cybernetics*59, pp. 33–40.MathSciNetMATHCrossRefGoogle Scholar - [7]Fukunaga, K., (1990),
*Introduction to Statistical Pattern Recognition*,*2nd edition*, Academic Press, San Diego.MATHGoogle Scholar - [8]Jeffreys, C. and T. Niznik, (1994), “Easing the Consious of the Guilty Net,”
*Computers and Operations Research*21, pp. 961–968.CrossRefGoogle Scholar - [9]Kohonen, T., (1995),
*Self-Organizing Maps*, Springer-Verlag, Berlin.Google Scholar - [10]Lawler, E. L., J. K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, (1985),
*The Travelling Salesman Problem: A Guided Tour of Combinatorial Optimization*, Wiley, Chichester.Google Scholar - [11]Papadimitriou, C. H., (1978), “The Euclidean Travelling Salesman Problem is NP-Complete”,
*Theoretical Computer Science*, 4, pp. 237–244.MathSciNetCrossRefGoogle Scholar - [12]Potvin, J.-Y., (1993), “The Travelling Salesman Problem: A Neural Network Perspective,”
*ORSA Journal on Computing*5, pp. 328–348.MATHGoogle Scholar - [13]Reinelt, G., (1991), “TSPLIB—A Travelling Salesman Problem Library”,
*ORSA Journal on Computing*3, pp. 376–384.Google Scholar - [14]Reinelt, G., (1994),
*The Travelling Salesman. Computational Solutions for TSP Applications*, Springer-Verlag, Berlin.Google Scholar - [15]Wong, Y., (1996), “A Comparative Study of the Kohonen Self-Organizing Map and the Elastic Net,”
*Computational Learning Theory and Natural Learning Systems*, 2, pp. 401–413.MATHGoogle Scholar