Solving maximum clique and independent set of graphs based on Hopfield network

Maximum clique and independent set problems are classical NP-full optimization problems, the solutions of which are difficult to obtain from conventional methods. Hopfield network in neural network, which simulates the partial functions of a human brain through the ultra-large scale parallel computation, has been proven to have potentials in solving these problems in a reasonable period of time. The main problem of this approach is the difficulty in defining an efficient energy function and the dynamic equation of motion for the Hopfield model. In this paper, we propose solutions to this problem by solving two typical problems in the coloring of graphs, the maximum clique and independent set, through our refined Hopfield network model. Both the mathematical model and the simulation algorithm are given here. It is found that the time complexity to obtain an optimal solution can approach one order of magnitude lower than the current available solutions.