Nonpositive Hopfield Neural Network with Self-Feedback and its Application to Maximum Clique Problems

A clique of an undirected graph G (V, E) with a vertex set V and an edge set E is a subset of V such that all pairs of vertices are connected by an edge in E. The maximum clique problem (MCP) is to find a clique of maximum size of the graph G. Figure 1(b) shows a maximum clique of the graph Figure 1(a) with 10 vertices and 21 edges. It is one of the first problems which have been proven to be NP-complete [1]. This problem is computationally intractable even to approximate with certain absolute performance bounds [2] [3]. Since the MCP is NP-complete for which the exact solution is difficult to obtain, different heuristic or approximation algorithms have been proposed. Shrivastava et al. [4] proposed a nonpositive Hopfield neural network (NHNN) with nonpositive synapses and zero thresholds and applied the NHNN to the graph optimization problems, for example, MCP, maximum independent set problem (MISP) and vertex cover problem (VCP). In 1996, Takefuji et al. proposed a maximum neural network (MNN) for solving the MCP [5]. Recently, Galán-Marín et al. showed that the parallel MNN algorithm did not always guarantee the descent of the energy function; therefore it can not guarantee the convergence to a clique of the graph [6]. In contrast, they proposed an optimal competitive Hopfield network model (OCHOM) for the MCP [6]. Their simulation results illustrated that the OCHOM was computationally superior to the parallel MNN in terms of both the solution quality and the computation time. In this paper, by adding a negative self-feedback to the NHNN and gradually removing this negative self-feedback, the proposed algorithm can help the NHNN escape from local minima and has powerful ability of searching the globally optimal or near-optimum solution for the MCP. The performance of the proposed algorithm is evaluated by simulating a number of instances.