Effective neural algorithms for the traveling salesman problem

Recently, several researchers have tried to solve the Traveling Salesman Problem (TSP) using the Hopfield model. Unfortunately, the results obtained so far are not good enough compared with the results obtained by conventional heuristic algorithms. This article presents two new neural algorithms. The first one uses Hopfield's neural network, and is a neural implementation of the Karp and Steele algorithm. Based on a generalized neural network, the second neural algorithm improves the first one by adaptively changing the neural network and thus the optimization function. On 40 TSP instances with random distance matrices, the Neural Algorithm 2 is better than Neural Algorithm 1 with confidence level α = 0.05, and much better than the 2-OPT (30 runs) and the Lin and Kernighan algorithm (30 runs). For random planar instances, Neural Algorithm 2 provides excellent initial solutions for iterative improvement algorithms, such as 2-OPT or Lin and Kernighan algorithm. For example, on 40 instances (each having 100 cities), the results of Neural Algorithm 2 improved by 2-OPT are better by at least 1.1% (with confidence level α = 0.0069) than the results produced by 2-OPT with 25 random initial solutions. If the results are postprocessed by the Lin and Kernighan algorithm, the Neural Algorithm 2 also outperforms the Lin and Kernighan algorithm with 20 random initial solutions (for each problem instance). Furthermore, the Neural Algorithm 2 scales up better than Lin and Kernighan algorithm (30 runs).