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An Efficient Multivalued Hopfield Network for the Traveling Salesman Problem

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Abstract

In this Letter we show that discrete multivalued Hopfield-type neural networks enable a relatively easy formulation of the Traveling Salesman Problem compared to the traditional Hopfield model. Thus, with the multivalued representation the network can be easily confined to feasible solutions, avoiding the need to tune any parameter. An investigation into the performance of the network has led us to define updating rules based on simple effective heuristic algorithms, a technique that can not be usually incorporated into standard Hopfield models. Simulation results for Euclidean Traveling Salesman Problems taken from the data library TSPLIB [11] indicate that this multivalued neural approach is superior to the best neural network currently reported for this problem.

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References

  1. Hopceld, J. J. and Tank, D. W.: Neural computation of decisions in optimization problems, Biol. Cybern. 52 (1985), 141–152.

    Google Scholar 

  2. Hopceld, J. J.: Neural Networks and physical systems with emergent collective computational abilities, Proc. Ac. Academy Sci. USA 79 (1982), 2554–2558.

    Google Scholar 

  3. Hopceld, J. J.: Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Ac. Academy Sci. USA 81 (1984), 3088–3092.

    Google Scholar 

  4. Glesner, M. and Pöchmüller, W.: Neurocomputers-An overview of neural Networks in VLSI. London: Chapman and Hall, 1994.

    Google Scholar 

  5. Shams, S. and Gaudiot, J. L.: Implementing regularly structured neural networks on the DREAM machine, IEEE Trans. Neural Networks 6 (1995), 407–421.

    Google Scholar 

  6. Farhat, N. H., Psaltis, D., Prata, A. and Peak, E.: Optical implementation of the Hopceld model, Appl. Opt. 4 (1985), 1469–1475.

    Google Scholar 

  7. Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G. and Shmoys, D. B.: The traveling salesman problem, New York: Wiley, 1985.

    Google Scholar 

  8. Reeves, C. R.: Modern heuristic techniques for combinatorial problems, Oxford: Blackwell, 1993.

    Google Scholar 

  9. Garey, M. R. and Johnson, D. S.: Computers and Intractability: A Guide to the theory of NP-Completeness, New York: W. H. Freeman, 1979.

    Google Scholar 

  10. Lin, S. and Kernighan, B.: An effective heuristic algorithm for the traveling salesman problem, Operations Research 21 (1973), 498–516.

    Google Scholar 

  11. Reinelt, G.: TSPLIB-a travelling salesman problem library, ORSA Journal on Computing 3 (1991), 376–384.

    Google Scholar 

  12. Reinelt, G.: The travelling salesman. Computational solutions for TSP applications, Berlin: Springer, 1994.

    Google Scholar 

  13. Aras, N., Oomen, B. J. and Altinel, I. K.: The Kohonen network incorporating explicit statistics and its application to the travelling salesman problem, Neural Networks 12 (1999), 1273–1284.

    Google Scholar 

  14. Burke, L. I. and Damany, P.: The guilty net for the travelling salesman problem, Computers and Operations Research 19 (1992), 255–265.

    Google Scholar 

  15. Kohonen, T.: Self-organizing maps, Berlin: Springer, 1995.

    Google Scholar 

  16. Gee, A. H. and Prager, R. W.: Polyhedral combinatorics and neural networks, Neural Computation 6(1) (1994), 161–180.

    Google Scholar 

  17. Smith, K.: An argument for abandoning the traveling salesman problem as a neuralnetwork benchmark, IEEE Trans on Neural Networks 7(6) (1996), 1542–1544.

    Google Scholar 

  18. Wilson, V. and Pawley, G. S.: On the stability of the TSP problem algorithm of Hopceld and Tank, Biological Cybernetics 58 (1988), 63–70.

    Google Scholar 

  19. Erdem, M. and Ozturk, Y.: A New Family of Multivalued Networks, Neural Networks 9(6) (1996), 979–989.

    Google Scholar 

  20. Peng, M., Gupta, N. K. and Armitage, A. A. F.: An investigation into the Improvement of Local Minima of the Hopceld Network, Neural Networks 9(7) (1996), 1241–1253.

    Google Scholar 

  21. Li, S. Z.: Improving Convergence and Solution Quality of Hopceld-Type Neural Networks with Augmented LagrangeMultipliers, IEEE Trans. on Neural Networks 7(6) (1996), 1507–1516.

    Google Scholar 

  22. Martín-Valdivia, M., Ruiz-Sepúlveda, A. and Triguero-Ruiz, F.: Improving local minima of Hopceld networks with augmented Lagrange multipliers for large scale TSPs, Neural Networks 13 (2000), 283–285.

    Google Scholar 

  23. Zeng, X. and Martinez, T.: A New Relaxation Procedure in the Hopceld Network for solving Optimization Problems, Neural Processing Letters 10 (1999), 211–222.

    Google Scholar 

  24. Johnson, D. S. and McGeoh, L. A.: The Traveling Salesman Problem: A Case Study in Local Optimization, In: E. H. Aarts and J. K. Lenstra (eds.), Local Search in Combinatorial Optimization, John Wiley and Sons: London, 1997, pp. 215–310.

    Google Scholar 

  25. Kirkpatrick, S.: Optimization by simulated annealing: Quantitative studies, J. Stat. Physics 34 (1984), 976-986.

    Google Scholar 

  26. Kirkpatrick, S., Gelatt, C. D. and Vecchi, M. P.: Optimization by Simulated Annealing, Science 220 (1983), 671–680.

    Google Scholar 

  27. Cerny, V.: A Thermodynamical Approach to the Travelling Salesman Problem: An Efccient Simulation Algorithm, J. Optimization Theory and Appl. 45 (1985), 41–51.

    Google Scholar 

  28. Croes, G. A.: A method for solving traveling salesman problems, Operations Research 6 (1958) 791–812.

    Google Scholar 

  29. Glover, F.: Future paths for integer programming and links to articcial intelligence, Computers & Ops. Res. 13 (1986), 533–549.

    Google Scholar 

  30. Mehta, S. and Fulop, L.: An analog Neural Network to Solve the Hamiltonian Cycle Problem, Neural Networks 6 (1993), 869–881.

    Google Scholar 

  31. Angéniol, B., Vaubois, G. and Texier, J.: Self-organizing feature maps and the Traveling Salesman problem, Neural Networks 1 (1988), 289–283.

    Google Scholar 

  32. Durbin, R. and Willshaw, D.: An analogue approach to the Traveling Salesman Problem using an elastic net method, Nature 326 (1987), 689–691.

    Google Scholar 

  33. Xu, X. and Tsai, W. T.: Effective neural algorithms for the travelling salesman problem, Neural Networks 4 (1991), 193–205.

    Google Scholar 

  34. Galán-Marín, G. and Muñoz-Pérez, J.: Design and Analysis of Maximum Hopceld Networks, IEEE Trans. on Neural Networks 12(2) (2001), 1–11.

    Google Scholar 

  35. Galán-Marín, G. and Muñoz-Pérez, J.: A new input-output function for binary Hopceld Neural Networks, Lecture Notes in Computer Science, Foundations and Tools for Neural Modeling, Springer Verlag, Berlin, (1999) vol. 1606, pp. 311–320.

    Google Scholar 

  36. Galán-Marín, G. and Muñoz-Pérez, J.: Design of a Neural Network for solving the Four-Color-Map Problem, Iberoamerican Journal of Artificial Intelligence 8 (1999), 6–17, (In Spanish).

    Google Scholar 

  37. Galán-Marín, G.: Recurrent Neural Networks for Combinatorial Optimization, Ph.D. dissertation, University of Málaga, Málaga, Spain, 2000 (In Spanish).

    Google Scholar 

  38. Galán-Marín, G. and Muñoz-Pérez, J.: Finding Near-Maximum Cliques with Competitive Hopceld Networks, Proc. of International ICSC Symposium on Neural Computation, Berlin, Germany (2000), pp. 62–69.

  39. Mérida-Casermeiro, E.: Recurrent Multivalued Neural Networks for Combinatorial Optimization and Pattern Recognition, Ph.D. dissertation, University of Málaga, Málaga, Spain, 2000 (In Spanish).

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Málaga, Campus de Teatinos s/n, 29071, Málaga, Spain

    E. Mérida-Casermeiro

  2. Department of Computer Science, University of Málaga, Campus de Teatinos s/n, 29071, Málaga, Spain

    J. Muñoz-Pérez

  3. Department of Electronics and Electromechanical Engineering, University of Extremadura, Escuela de Ingenierías Industriales, Avda. de Elvas s/n, 06071, Badajoz, Spain

    G. Galán-Marín

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  1. E. Mérida-Casermeiro
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  2. G. Galán-Marín
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  3. J. Muñoz-Pérez
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Mérida-Casermeiro, E., Galán-Marín, G. & Muñoz-Pérez, J. An Efficient Multivalued Hopfield Network for the Traveling Salesman Problem. Neural Processing Letters 14, 203–216 (2001). https://doi.org/10.1023/A:1012751230791

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