Discrete Optimization The graph coloring problem: A neuronal network approach q Pedro M. Talava ´n a , Javier Ya ´n ˜ez b, * a Ministry of Economy and Finance, Los Madrazo 38, 28014 Madrid, Spain b Department of Statistics and Operations Research, Complutense University, 28040 Madrid, Spain Received 12 July 2006; accepted 16 August 2007 Available online 7 September 2007 Abstract Solution of an optimization problem with linear constraints through the continuous Hopfield network (CHN) is based on an energy or Lyapunov function that decreases as the system evolves until a local minimum value is attained. This approach is extended in to optimization problems with quadratic constraints. As a particular case, the graph coloring problem (GCP) is analyzed. The mapping procedure and an appropriate parameter-setting procedure are detailed. To test the theoretical results, some computational experiments solving the GCP are shown. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Artificial neural networks; Hopfield network; Graph coloring problem 1. Introduction The continuous Hopfield network (CHN) was introduced by Hopfield and Tank [9] to solve the traveling salesman problem (TSP) and other combinatorial problems. Although some stimulating results were obtained when the parameters of the CHN were appropriately chosen manually, the feasibility of the equilibrium points of the differential equations of the CHN cannot be assured for the general case, and thus this was not a useful heuristic tool for solving such problems. We previously proposed a parameter setting for the TSP so that the feasibility of the equilibrium points of the CHN was guaranteed [14,15]. We then introduced a new energy function [16] that generalizes the function proposed by Hopfield and Tank. With this generalized energy function, the generalized quadratic knapsack problem, i.e., 0–1 problems with a quadratic objective function and linear constraints, which includes the TSP as a particular case, was solved by the CHN. In all the computational experiments carried out, valid solu- tions for the associated combinatorial problems were always obtained. This property is not a trivial task for some combinatorial optimization problems. 0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.08.034 q This research was supported by DGICYT (National Grant No. MTM2005-08982-C04-01). * Corresponding author. Tel.: +34 91 3944522; fax: +34 91 3944606. E-mail addresses: pedro.martinez@meh.es (P.M. Talava ´n), jayage@mat.ucm.es (J. Ya ´n ˜ ez). Available online at www.sciencedirect.com European Journal of Operational Research 191 (2008) 100–111 www.elsevier.com/locate/ejor