THE TRAVELING SALESMAN PROBLEM: A NEURAL NETWORK PERSPECTIVE

This paper surveys the "neurally" inspired problem-solving approaches to the traveling salesman problem, namely, the Hopfield-Tank network, the elastic net and the self-organizing map. The latest achievements in the neural network domain are reported and numerical comparisons are provided with the classical solution approaches of operations research. An extensive bibliography with more than one hundred references is also included. (A)

[1]  D. J. Burr,et al.  An improved elastic net method for the traveling salesman problem , 1988, IEEE 1988 International Conference on Neural Networks.

[2]  Jon Jouis Bentley,et al.  Fast Algorithms for Geometric Traveling Salesman Problems , 1992, INFORMS J. Comput..

[3]  Y. Akiyama,et al.  Combinatorial optimization with Gaussian machines , 1989, International 1989 Joint Conference on Neural Networks.

[4]  Yasuo Matsuyama,et al.  Self-organizing neural networks and various euclidean traveling salesman problems , 1992, Systems and Computers in Japan.

[5]  Fred W. Glover,et al.  Tabu Search - Part I , 1989, INFORMS J. Comput..

[6]  Yasuo Matsuyama,et al.  Competitive self-organization and combinatorial optimization: applications to traveling salesman problem , 1990, 1990 IJCNN International Joint Conference on Neural Networks.

[7]  David S. Johnson,et al.  Local Optimization and the Traveling Salesman Problem , 1990, ICALP.

[8]  David E. van den Bout,et al.  A traveling salesman objective function that works , 1988, IEEE 1988 International Conference on Neural Networks.

[9]  D. E. Van den Bout,et al.  TInMANN: the integer Markovian artificial neural network , 1989, International 1989 Joint Conference on Neural Networks.

[10]  Geoffrey E. Hinton,et al.  Learning and relearning in Boltzmann machines , 1986 .

[11]  Shigeo Abe.  Global convergence and suppression of spurious states of the Hopfield neural networks , 1991, [Proceedings] 1991 IEEE International Joint Conference on Neural Networks.

[12]  W. I. Clement,et al.  Synaptic Strengths For Neural Simulation Of The Traveling Salesman Problem , 1988, Defense, Security, and Sensing.

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

[14]  P. Bozovsky.  Discrete Hopfield model with graded response (analysis and applications) , 1990, 1990 IJCNN International Joint Conference on Neural Networks.

[15]  Emile H. L. Aarts,et al.  Simulated annealing and Boltzmann machines - a stochastic approach to combinatorial optimization and neural computing , 1990, Wiley-Interscience series in discrete mathematics and optimization.

[16]  Liu Ze-Min,et al.  Determination of the parameters in a modified Hopfield-Tank model (2) on solving TSP , 1991, [Proceedings] 1991 IEEE International Joint Conference on Neural Networks.

[17]  James P. Ignizio,et al.  Neural networks and operations research: An overview , 1992, Comput. Oper. Res..

[18]  Martin W. Simmen.  Parameter Sensitivity of the Elastic Net Approach to the Traveling Salesman Problem , 1991, Neural Computation.

[19]  Kevin Knight,et al.  Connectionist ideas and algorithms , 1990, CACM.

[20]  K. Schulten,et al.  Kohonen's self-organizing maps: exploring their computational capabilities , 1988, IEEE 1988 International Conference on Neural Networks.

[21]  Richard Durbin,et al.  An analogue approach to the travelling salesman problem using an elastic net method , 1987, Nature.

[22]  X. Xu,et al.  Effective neural algorithms for the traveling salesman problem , 1991, Neural Networks.

[23]  Sanjit K. Mitra,et al.  Alternative networks for solving the traveling salesman problem and the list-matching problem , 1988, IEEE 1988 International Conference on Neural Networks.

[24]  Katsunori Shimohara,et al.  An approach for solving dynamic TSPs using neural networks , 1991, [Proceedings] 1991 IEEE International Joint Conference on Neural Networks.

[25]  Ramesh Sharda,et al.  Impacts of recent computer advances on operations research , 1989 .

[26]  N. Biggs.  THE TRAVELING SALESMAN PROBLEM A Guided Tour of Combinatorial Optimization , 1986 .

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

[28]  R. Palmer,et al.  Introduction to the theory of neural computation , 1994, The advanced book program.

[29]  E. Aarts,et al.  Boltzmann machines for travelling salesman problems , 1989 .

[30]  Anders Krogh,et al.  Introduction to the theory of neural computation , 1994, The advanced book program.

[31]  Daniel J. Rosenkrantz,et al.  An Analysis of Several Heuristics for the Traveling Salesman Problem , 1977, SIAM J. Comput..

[32]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[33]  Sweet.  Determination of parameters in a Hopfield/Tank computational network , 1988 .

[34]  Won Don Lee,et al.  Parallel mean field annealing neural network for solving traveling salesman problem , 1992, [Proceedings 1992] IJCNN International Joint Conference on Neural Networks.

[35]  Aart Joppe,et al.  A neural network for solving the travelling salesman problem on the basis of city adjacency in the tour , 1990, 1990 IJCNN International Joint Conference on Neural Networks.

[36]  Giovanni Rinaldi,et al.  Facet identification for the symmetric traveling salesman polytope , 1990, Math. Program..

[37]  G. G. Coghill,et al.  Genetic breeding of control parameters for the Hopfield/Tank neural net , 1992, [Proceedings 1992] IJCNN International Joint Conference on Neural Networks.

[38]  Laura I. Burke,et al.  Neural methods for the traveling salesman problem: Insights from operations research , 1994, Neural Networks.

[39]  Fred Glover,et al.  Tabu Search - Part II , 1989, INFORMS J. Comput..

[40]  Sheng-De Wang,et al.  Hopfield nets with time-varying energy functions for solving the travelling salesman problem , 1991, [Proceedings] 1991 IEEE International Joint Conference on Neural Networks.

[41]  Yasuo Matsuyama,et al.  Self-organization via competition, cooperation and categorization applied to extended vehicle routing problems , 1991, IJCNN-91-Seattle International Joint Conference on Neural Networks.

[42]  Behrooz Kamgar-Parsi,et al.  On problem solving with Hopfield neural networks , 1990, International 1989 Joint Conference on Neural Networks.

[43]  H. Szu,et al.  Fast TSP algorithm based on binary neuron output and analog neuron input using the zero-diagonal interconnect matrix and necessary and sufficient constraints of the permutation matrix , 1988, IEEE 1988 International Conference on Neural Networks.

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

[45]  Derek M. Wells,et al.  Solving degenerate optimization problems using networks of neural oscillators , 1992, Neural Networks.

[46]  Alan L. Yuille,et al.  Generalized Deformable Models, Statistical Physics, and Matching Problems , 1990, Neural Computation.

[47]  G. Laporte.  The traveling salesman problem: An overview of exact and approximate algorithms , 1992 .

[48]  Mandell Bellmore,et al.  Transformation of Multisalesman Problem to the Standard Traveling Salesman Problem , 1974, JACM.

[49]  H. Szu,et al.  Solving large-scale optimization problems by divide-and-conquer neural networks , 1989, International 1989 Joint Conference on Neural Networks.

[50]  Shashank K. Mehta,et al.  A neural algorithm to solve the Hamiltonian cycle problem , 1990, 1990 IJCNN International Joint Conference on Neural Networks.

[51]  Liu Ze-Min,et al.  Parameters rules of the Hopfield/Tank model on solving TSP , 1992, [Proceedings 1992] IJCNN International Joint Conference on Neural Networks.

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

[53]  William B. Levy,et al.  Determination of parameters in a Hopfield/Tank computational network , 1988, IEEE 1988 International Conference on Neural Networks.

[54]  B. Fritzke,et al.  FLEXMAP-a neural network for the traveling salesman problem with linear time and space complexity , 1991, [Proceedings] 1991 IEEE International Joint Conference on Neural Networks.

[55]  Teuvo Kohonen,et al.  Self-Organization and Associative Memory , 1988 .

[56]  Gilbert Laporte,et al.  New Insertion and Postoptimization Procedures for the Traveling Salesman Problem , 1992, Oper. Res..

[57]  Chee-Kit Looi,et al.  Neural network methods in combinatorial optimization , 1992, Comput. Oper. Res..

[58]  Geoffrey E. Hinton,et al.  Proceedings of the 1988 Connectionist Models Summer School , 1989 .

[59]  D J Willshaw,et al.  A marker induction mechanism for the establishment of ordered neural mappings: its application to the retinotectal problem. , 1979, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

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

[61]  Bernard Angéniol,et al.  Self-organizing feature maps and the travelling salesman problem , 1988, Neural Networks.

[62]  Carsten Peterson,et al.  A Mean Field Theory Learning Algorithm for Neural Networks , 1987, Complex Syst..

[63]  David E. van den Bout,et al.  Encoding logical constraints into neural network cost functions , 1990, 1990 IJCNN International Joint Conference on Neural Networks.

[64]  Richard Szeliski,et al.  An Analysis of the Elastic Net Approach to the Traveling Salesman Problem , 1989, Neural Computation.

[65]  R. C. Mann,et al.  An extension of the Hopfield-Tank model for solution of the multiple traveling salesmen problem , 1988, IEEE 1988 International Conference on Neural Networks.

[66]  A. B. Kahng.  Traveling salesman heuristics and embedding dimension in the Hopfield model , 1989, International 1989 Joint Conference on Neural Networks.

[67]  M.C.S. Boeres,et al.  A faster elastic-net algorithm for the traveling salesman problem , 1992, [Proceedings 1992] IJCNN International Joint Conference on Neural Networks.

[68]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[69]  Li-Chen Fu,et al.  Ineffectiveness in solving combinatorial optimization problems using a Hopfield network: a new perspective from aliasing effect , 1990, IJCNN.

[70]  Jean-Yves Potvin,et al.  Neural networks for automated vehicle dispatching , 1992, Comput. Oper. Res..

[71]  Giovanni Rinaldi,et al.  A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems , 1991, SIAM Rev..

[72]  S. Abe.  Theories on the Hopfield neural networks , 1989, International 1989 Joint Conference on Neural Networks.

[73]  Mahesan Niranjan,et al.  A theoretical investigation into the performance of the Hopfield model , 1990, IEEE Trans. Neural Networks.

[74]  Laura I. Burke,et al.  The guilty net for the traveling salesman problem , 1992, Comput. Oper. Res..

[75]  Petar D. Simic,et al.  Statistical mechanics as the underlying theory of ‘elastic’ and ‘neural’ optimisations , 1990 .

[76]  Shen Lin.  Computer solutions of the traveling salesman problem , 1965 .

[77]  Behrooz Shirazi,et al.  Critical analysis of applying Hopfield neural net model to optimization problems , 1989, Conference Proceedings., IEEE International Conference on Systems, Man and Cybernetics.

[78]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..

[79]  Emile H. L. Aarts,et al.  Boltzmann Machines and their Applications , 1987, PARLE.

[80]  David M Stein.  SCHEDULING DIAL-A-RIDE TRANSPORTATION SYSTEMS: AN ASYMPTOTIC APPROACH , 1977 .

[81]  Samuel J. Raff,et al.  Routing and scheduling of vehicles and crews : The state of the art , 1983, Comput. Oper. Res..

[82]  G. J. Hueter.  Solution of the traveling salesman problem with an adaptive ring , 1988, IEEE 1988 International Conference on Neural Networks.