Application of the edge of chaos in combinatorial optimization*

Abstract

Many problems in science, engineering and real life are related to the combinatorial optimization. However, many combinatorial optimization problems belong to a class of the NP-hard problems, and their globally optimal solutions are usually difficult to solve. Therefore, great attention has been attracted to the algorithms of searching the globally optimal solution or near-optimal solution for the combinatorial optimization problems. As a typical combinatorial optimization problem, the traveling salesman problem (TSP) often serves as a touchstone for novel approaches. It has been found that natural systems, particularly brain nervous systems, work at the critical region between order and disorder, namely, on the edge of chaos. In this work, an algorithm for the combinatorial optimization problems is proposed based on the neural networks on the edge of chaos (ECNN). The algorithm is then applied to TSPs of 10 cities, 21 cities, 48 cities and 70 cities. The results show that ECNN algorithm has strong ability to drive the networks away from local minimums. Compared with the transiently chaotic neural network (TCNN), the stochastic chaotic neural network (SCNN) algorithms and other optimization algorithms, much higher rates of globally optimal solutions and near-optimal solutions are obtained with ECNN algorithm. To conclude, our algorithm provides an effective way for solving the combinatorial optimization problems.