Modelling competitive Hopfield networks for the maximum clique problem

Authors:
G. Galán-Marín;E. Mérida-Casermeiro;J. Muñoz-Pérez
Affiliations:
Department of Electronics and Electromechanical Engineering, University of Extremadura, Escuela de Ingenierías Industriales, Avda. de Elvas s/n, 06071 Badajoz, Spain;Department of Applied Mathematics, University of Málaga, Málaga, Spain;Department of Computer Science, University of Málaga, Málaga, Spain
Venue:
Computers and Operations Research
Year:
2003

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Abstract

The maximum clique problem (MCP) is a classic graph optimization problem with many real-world applications. This problem is NP-complete and computationally intractable even to approximate with certain absolute performance bounds. In this paper, we present the design of a new discrete competitive Hopfield neural network for finding near-optimum solutions for the MCP. Other competitive Hopfield-style networks have been proposed to solve the MCP. However, recent results have shown that these models can sometimes lead to inaccurate results and oscillatory behaviors in the convergence process. Thus, the network sometimes does not converge to cliques of the considered graph, where this error usually increases with the size and the density of the graph. In contrast, we propose in this paper appropriate dynamics for a binary competitive Hop-field network in order to always generate local/global minimum points corresponding to maximal/maximun cliques of the considered graph. Moreover, an optimal modelling of the network is developed, resulting in a fast two-level winner-take-all scheme. Extensive simulation runs show that our network performs better than the other competitive neural approaches in terms of the solution quality and the computation time.