Abstract
The map-coloring problem is a well known combinatorial optimization problem which frequently appears in mathematics, graph theory and artificial intelligence. This paper presents a study into the performance of some binary Hopfield networks with discrete dynamics for this classic problem. A number of instances have been simulated to demonstrate that only the proposed binary model provides optimal solutions. In addition, for large-scale maps an algorithm is presented to improve the local minima of the network by solving gradually growing submaps of the considered map. Simulation results for several n-region 4-color maps showed that the proposed neural algorithm converged to a correct colouring from at least 90% of initial states without the fine-tuning of parameters required in another Hopfield models.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Saaty, T., Hainen, P.: The four color theorem: Assault and Conquest. McGraw-Hill, New York (1977)
Appel, K., Haken, W.: The solution of the four-color-map problem. Scientific American, 108–121 (Oct. 1977)
Takefuji, Y., Lee, K.C.: Artificial neural networks for four-colouring map problems and K-colorability problems. IEEE Trans. Circuits Syst. 38, 326–333 (1991)
Funabiki, N., Takenaka, Y., Nishikawa, S.: A maximum neural network approach for N-queens problem. Biol. Cybern. 76, 251–255 (1997)
Funabiki, N., Takefuji, Y., Lee, K.C.: A Neural Network Model for Finding a Near-Maximum Clique. J. of Parallel and Distributed Computing 14, 340–344 (1992)
Lee, K.C., Funabiki, N., Takefuji, Y.: A Parallel Improvement Algorithm for the Bipartite Subgraph Problem. IEEE Trans. Neural Networks 3, 139–145 (1992)
Galán-Marín, G., Muñoz-Pérez, J.: Design and Analysis of Maximum Hopfield Networks. IEEE Transactions on Neural Networks 12, 329–339 (2001)
Galán-Marín, G., Mérida-Casermeiro, E., Muñoz-Pérez, J.: Modelling competitive Hopfield networks for the maximum clique problem. Computers & Operations Research 30, 603–624 (2003)
Wang, L.: Discrete-time convergence theory and updating rules for neural networks with energy functions. IEEE Trans. Neural Networks 8, 445–447 (1997)
Tateishi, M., Tamura, S.: Comments on ’Artificial neural networks for four-colouring map problems and K-colorability problems’. IEEE Trans. Circuits Syst. I: Fundamental Theory Applicat. 41, 248–249 (1994)
Galán-Marín, G., Muñoz-Pérez, J.: A new input-output function for binary Hopfield Neural Networks. In: Mira, J. (ed.) IWANN 1999. LNCS, vol. 1606, pp. 311–320. Springer, Heidelberg (1999)
Wang, J., Tang, Z.: An improved optimal competitive Hopfield network for bipartite subgraph problems. Neurocomputing 61, 413–419 (2004)
Joya, G., Atencia, M.A., Sandoval, F.: Hopfield neural networks for optimization: study of the different dynamics. Neurocomputing 43, 219–237 (2002)
Dahl, E.D.: Neural Network algorithm for an NP-Complete problem: Map and graph coloring. In: Proc. First Int. Joint Conf. on Neural Networks III, pp. 113–120 (1987)
Sun, Y.: A generalized updating rule for modified Hopfield neural network for quadratic optimization. Neurocomputing 19, 133–143 (1998)
Peng, M., Gupta, N.K., Armitage, A.F.: An investigation into the improvement of local minima of the Hopfield network. Neural Networks 9, 1241–1253 (1996)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Galán-Marín, G., Mérida-Casermeiro, E., López-Rodríguez, D., Ortiz-de-Lazcano-Lobato, J.M. (2007). A Study into the Improvement of Binary Hopfield Networks for Map Coloring. In: Beliczynski, B., Dzielinski, A., Iwanowski, M., Ribeiro, B. (eds) Adaptive and Natural Computing Algorithms. ICANNGA 2007. Lecture Notes in Computer Science, vol 4432. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71629-7_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-71629-7_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71590-0
Online ISBN: 978-3-540-71629-7
eBook Packages: Computer ScienceComputer Science (R0)