Soft Computing

, Volume 21, Issue 14, pp 3891–3905 | Cite as

Improving the Hopfield model performance when applied to the traveling salesman problem

A divide-and-conquer scheme
  • Lucas García
  • Pedro M. Talaván
  • Javier Yáñez
Methodologies and Application


The continuous Hopfield network (CHN) can be used to solve, among other combinatorial optimization problems, the traveling salesman problem (TSP). In order to improve the performance of this heuristic technique, a divide-and-conquer strategy based on two phases is proposed. The first phase involves linking cities with the most neighbors to define a set of chains of cities and, secondly, to join these with isolated cities to define the final tour. Both problems are solved by mapping the two TSPs onto their respective CHNs. The associated parameter-setting procedures are deduced; these procedures ensure the feasibility of the obtained tours, and the quality of the solutions is compared with the pure CHN approach using some traveling salesman problem library (TSPLIB) instances. By means of this strategy, solving TSP instances with sizes of up to 13,509 cities are allowed with the computational resources we had available. Finally, the new divide-and-conquer procedure is improved by tuning the parameter which controls the first phase.


Artificial neural networks Hopfield network Traveling salesman problem Divide-and-conquer 



We thank the two anonymous referees for their interesting suggestions. This research has been partially supported by the Government of Spain, Grant TIN2012-32482, and by the local Government of Madrid, Grant S2013/ICE-2845 (CASI-CAM).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest.


  1. Di Marco M, Forti M, Grazzini M, Pancioni L (2014) Necessary and sufficient condition for multistability of neural networks evolving on a closed hypercube. Neural Netw 54:38–48CrossRefzbMATHGoogle Scholar
  2. Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci 81(10):3088–3092CrossRefGoogle Scholar
  3. Hopfield JJ, Tank DW (1985) “Neural” computation of decisions in optimization problems. Biol Cybern 52(3):141–152zbMATHGoogle Scholar
  4. Jolai F, Ghanbari A (2010) Integrating data transformation techniques with Hopfield neural networks for solving travelling salesman problem. Expert Syst Appl 37(7):5331–5335CrossRefGoogle Scholar
  5. Lawler EL (1985) The traveling salesman problem: a guided tour of combinatorial optimization. In: Wiley-Interscience series in discrete mathematics. Wiley, New YorkGoogle Scholar
  6. Mérida-Casermeiro E, Galán-Marín G, Munoz-Perez J (2001) An efficient multivalued Hopfield network for the traveling salesman problem. Neural Process Lett 14(3):203–216CrossRefzbMATHGoogle Scholar
  7. Papadimitriou CH (1977) The Euclidean travelling salesman problem is NP-complete. Theor Comput Sci 4(3):237–244CrossRefzbMATHGoogle Scholar
  8. Reinelt G (1991) TSPLIB. A traveling salesman problem library. ORSA J Comput 3(4):376–384CrossRefzbMATHGoogle Scholar
  9. Reinelt G (1994) The traveling salesman: computational solutions for TSP applications. Springer, New YorkGoogle Scholar
  10. Salcedo-Sanz S, Ortiz-García EG, Pérez-Bellido ÁM, Portilla-Figueras A, López-Ferreras F (2009) On the performance of the LP-guided Hopfield network-genetic algorithm. Comput Oper Res 36(7):2210–2216MathSciNetCrossRefzbMATHGoogle Scholar
  11. Schmidhuber J (2015) Deep learning in neural networks: an overview. Neural Netw 61:85–117CrossRefGoogle Scholar
  12. Suh T, Esat II (1998) Solving large scale combinatorial optimisation problems based on a divide and conquer strategy. Neural Comput Appl 7(2):166–179Google Scholar
  13. Talaván PM, Yáñez J (2002) Parameter setting of the Hopfield network applied to TSP. Neural Netw 15(3):363–373CrossRefGoogle Scholar
  14. Talaván PM, Yáñez J (2005) A continuous Hopfield network equilibrium points algorithm. Comput Oper Res 32(8):2179–2196MathSciNetCrossRefzbMATHGoogle Scholar
  15. Talaván PM, Yáñez J (2006) The generalized quadratic knapsack problem. A neuronal network approach. Neural Netw 19(4):416–428CrossRefzbMATHGoogle Scholar
  16. Tan K, Tang H, Ge S (2005) On parameter settings of Hopfield networks applied to traveling salesman problems. IEEE Trans Circuits Syst I Regul Pap 52(5):994–1002MathSciNetCrossRefGoogle Scholar
  17. Tang H, Tan KC, Zhang Y (2007) Neural networks: computational models and applications. Studies in Computational Intelligence, vol 53, Springer, Berlin, HeidelbergGoogle Scholar
  18. Wang RL, Tang Z, Cao QP (2002) A learning method in Hopfield neural network for combinatorial optimization problem. Neurocomputing 48(1):1021–1024CrossRefzbMATHGoogle Scholar
  19. Wen U-P, Lan K-M, Shih H-S (2009) A review of Hopfield neural networks for solving mathematical programming problems. Eur J Oper Res 198(3):675–687MathSciNetCrossRefzbMATHGoogle Scholar
  20. Wilson G, Pawley G (1988) On the stability of the travelling salesman problem algorithm of Hopfield and tank. Biol Cybern 58(1):63–70MathSciNetCrossRefzbMATHGoogle Scholar
  21. Woeginger GJ (2003) Exact algorithms for NP-hard problems: a survey. In: Combinatorial Optimization Eureka, You Shrink!. Springer, New York, pp 185–207Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Universidad Complutense de MadridMadridSpain
  2. 2.Instituto Nacional de EstadísticaMadridSpain

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