Chaotic Potts Spin Model for Combinatorial Optimization Problems

In this paper we first show some of the bifurcation properties of Potts mean-field-theory annealing applied to traveling salesman problems. Due to these bifurcation properties, this approach, in general, produces non-optimal and non-unique solutions. As an alternative approach, we propose a nonequilibrium version of the Potts spin neural network, called chaotic Potts spin (CPS). CPS has several parameters, and bifurcations over each parameter are investigated. Next, experimental results are shown comparing CPS with several related approaches. CPS is good at obtaining optimal solutions for small-scale problems and semi-optimal solutions for relatively large-scale problems. We also describe a couple of CPS modifications: CPS with a heuristic method and CPS with a "chaotic annealing" method. These modified algorithms can produce even better CPS solutions. Copyright 1997 Elsevier Science Ltd.

[1]  K. Aihara,et al.  Chaotic neural networks , 1990 .

[2]  Kazuyuki Aihara,et al.  Chaotic simulated annealing by a neural network model with transient chaos , 1995, Neural Networks.

[3]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[4]  Brian W. Kernighan,et al.  An Effective Heuristic Algorithm for the Traveling-Salesman Problem , 1973, Oper. Res..

[5]  Carsten Peterson,et al.  A New Method for Mapping Optimization Problems Onto Neural Networks , 1989, Int. J. Neural Syst..

[6]  Sato,et al.  Bifurcations in mean-field-theory annealing. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  F. Y. Wu.  The Potts model , 1982 .

[8]  Carsten Peterson,et al.  Neural Networks and NP-complete Optimization Problems; A Performance Study on the Graph Bisection Problem , 1988, Complex Syst..

[9]  Masa-aki Sato,et al.  Bifurcations in mean-eld-theory annealing , 1996 .

[10]  M. Čadek.  Singularities and groups in bifurcation theory, volume I , 1990 .

[11]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[12]  Naoki Kurita,et al.  On the Hopfield Neural Networks and Mean Field Theory , 1996, Neural Networks.

[13]  Carsten Peterson,et al.  Parallel Distributed Approaches to Combinatorial Optimization: Benchmark Studies on Traveling Salesman Problem , 1990, Neural Computation.

[14]  Wesley E. Snyder,et al.  Optimization by Mean Field Annealing , 1988, NIPS.

[15]  KIICHI URAHAMA,et al.  A Gradient System Solution To Potts Mean Field Equations And Its Electronic Implementation , 1993, Int. J. Neural Syst..

[16]  Geoffrey E. Hinton,et al.  A Learning Algorithm for Boltzmann Machines , 1985, Cogn. Sci..

[17]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[18]  Hiroshi Nozawa,et al.  A neural network model as a globally coupled map and applications based on chaos. , 1992, Chaos.