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

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## Abstract

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.

## Keywords

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

### Acknowledgments

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.

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