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We propose a novel approach for combinatorial optimization problems. For solving the traveling salesman problems, we combine chaotic neurodynamics with heuristic algorithm. We select the heuristic algorithm of 2-opt as a basic part, because it is well understood that this simple algorithm is very effective for the traveling salesman problems. Although the conventional approaches with chaotic neurodynamics were only applied to such very small problems as 10 cities, our method exhibits higher performance for larger size problems with the order of 10 2 . [S0031-9007(97)04059-3] PACS numbers: 82.20.Mj, 02.60.Pn, 05.45. + b, 87.10. + e Various methods are proposed for solving the traveling salesman problem (TSP) which is one of the typical NP (nondeterministic polynomial)-hard combinatorial optimization problems. One of the new approaches, or modern heuristics, is based upon artificial neural networks. The basic concept of this approach was proposed by Hopfield and Tank [1], who applied an artificial neural network with symmetric mutual connections to the TSP, which has a kind of gradient descent dynamics, namely, a decreasing property of the computational energy function. Although this approach is very attractive from the viewpoint of an application of artificial neural networks, the Hopfield-Tank neural network has a notorious local minimum problem. In order to solve such a difficult problem, a new approach using the chaotic neural network [2,3] has been proposed [4-6]. The chaotic dynamics has several particular properties. One of them, self-similarity, is that attractors of chaotic dynamical systems usually have fractal structures. Therefore, chaotic search is expected to be efficient because the chaotic dynamics searches solutions of the TSP only along such a fractal structure with zero Lebesgue measure in the state space, if the optimum solution is located in the searching region.However, these methods based on the recurrent neural networks have two serious problems. First, the HopfieldTank neural network, which provides the basic framework of the approach with chaotic dynamics, requires n 3 n mutual connections where n is the number of neurons. In the case of solving an N-city TSP, the number of neurons n is N 2 [1]. Therefore, the number of mutual connections becomes n 2 N 4 . If the number of cities N increases, the number of mutual connections becomes huge, consequently calculation gets difficult. Second, constructing a closed feasible tour (which is a constraint in the case of solving the TSP and translated as starting from a city, visiting each city exactly once, and returning to the starting city) is not easy in this approach because a closed tour is realized only by firing patterns of the neural networks that satisfy the constraints, namely, the number of firing neurons must be 1 in each row and each column. If the state of neural networks does not satisfy the constraint term, it cannot form even a closed feasible tour.As another conventional approach for solving the TSP, several heuristic methods have b...

In this Letter, we propose a framework to transform a complex network to a time series. The transformation from complex networks to time series is realized by the classical multidimensional scaling. Applying the transformation method to a model proposed by Watts and Strogatz [Nature (London) 393, 440 (1998)], we show that ring lattices are transformed to periodic time series, small-world networks to noisy periodic time series, and random networks to random time series. We also show that these relationships are analytically held by using the circulant-matrix theory and the perturbation theory of linear operators. The results are generalized to several high-dimensional lattices.