Skip to main content
SpringerLink
Log in

Neural networks and graph theory

  • Published:
Science in China Series F: Information Sciences Aims and scope Submit manuscript

Abstract

The relationships between artificial neural networks and graph theory are considered in detail. The applications of artificial neural networks to many difficult problems of graph theory, especially NP-complete problems, and the applications of graph theory to artificial neural networks are discussed. For example graph theory is used to study the pattern classification problem on the discrete type feedforward neural networks, and the stability analysis of feedback artificial neural networks etc.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Peretto, P., An Introduction to the Modeling of Neural Networks, London: Cambridge University Press, 1992.

    MATH  Google Scholar 

  2. Gallant, S. I., Neural Network Learning and Expert Systems, Cambridge: The MIT Press, 1993.

    MATH  Google Scholar 

  3. Bondy, J. A., Murty, U. S. R., Graph Theory with Application, London, Basingtoke, New York: The MacMillan Press LTD, 1976.

    Google Scholar 

  4. Hopfield, J. J., Tank, D. W., Neural computation of dicisions optimization problem, Biol. Cybernetics, 1985, 52: 141–152.

    MATH  MathSciNet  Google Scholar 

  5. Gibbons, A., Algorithmic Graph Theory, Cambridge, London, New York: Cambridge University Press, 1985.

    MATH  Google Scholar 

  6. Dahl, E. D., Neural network algorithm for an NP-complete problem: map and graph coloring, Proc. IEEE First International Conference on Neural Networks, Vol. III, June, San Diego, New York: IEEE, 1987, 113–120.

    Google Scholar 

  7. Xu, Jin, Zhang Junvin, Bao Zheng, Coloring alogorithm of graph theory based on Hopfield neural network, Acta Electronica Sinica, 1996, 24(10): 8–13.

    Google Scholar 

  8. Ma Songde, Artificial neural networks and optimization, Proceedings of the First Chinese Conference on Articifial Neural Networks, Beijing (in Chinese), Beijing: China Neural Network Society, 1990, 22–28.

    Google Scholar 

  9. Chen Guoliang, Optimal problems based on ANN, Proceedings of the First Chinese Conference on Articifial Neural Networks, Beijing (in Chinese), Beijing: China Neural Network Society, 1990, 122–129.

    Google Scholar 

  10. Liang Weufa et al., Some algorithms of hard graph theory problems based on artificial neural networks, Proceedings of the First Chinese Conference on Articifial Neural Networks, Beijing (in Chinese), Beijing: China Neural Network Society, 1990, 924–927.

    Google Scholar 

  11. Xu Jin, Zhang, J. Y., Bao, Z., A new isomorphic algorithm based on Hopfield neural networks, Journal of Electronics, 1996, 18: 116–121.

    Google Scholar 

  12. Xu Jin, Self-complementary Graph Theory with Applications: Chapter 6, Xi'an: Xidian University Press, 1999.

    Google Scholar 

  13. Bron, C., Kerbosch, J., Finging all cliques of an undirected graph, Commun. Ass. Mach., 1973, 16: 575–577.

    MATH  Google Scholar 

  14. Gendreau, M., Picard, J. C., Zubieta, L., An efficient implicit enumeration algorithm for the maximum clique problem, Lecture Notes in Economics and Mathematical Systems (eds. Kurzhanski, A. et al.), New York: Springer-Verlag, 1988, 304: 79–91.

    Google Scholar 

  15. Mitten, L. G., Branch and bound method: general formulation and properties, Ops. Res., 1970, 18: 24–34.

    MATH  MathSciNet  Google Scholar 

  16. Robson, J. M., Algorithms for maximum independent sets, J. Algorithm, 1986, 7: 425–440.

    Article  MATH  MathSciNet  Google Scholar 

  17. Tarjan, R. E., Trojanowski, A. E., Finding a maximum independent set, SIAM JL Comput., 1977, 6: 537–546.

    Article  MATH  MathSciNet  Google Scholar 

  18. Balas, E., Yu, C. S., Finging the maximum clique in an arbitrary graph, SIAM JL Comput., 1986, 15: 1054–1068.

    Article  MATH  MathSciNet  Google Scholar 

  19. Panos, M. P., Gregory, P. R., A brance and bound algorithm for the maximum clique problem, Computers Ops. Res., 1992, 19(5): 363–375.

    Article  MATH  Google Scholar 

  20. Zhang, J. Y., Xu Jin, Bao, Z., A new algorithm of the maximum clique and the maximum independent set problems based on Hopfield neural network, Journal of Electronics, 1996, 18(sup): 121–127.

    Google Scholar 

  21. Booth, K. S., Lueker, G. S., Testing the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms, J. Comput. Syst. Sci., 1976, 13: 335–379.

    MATH  MathSciNet  Google Scholar 

  22. Hopcroft, J. E., Tarjan, R. E., Efficient planarity testing of graphs,J. Assoc. Comput. Mach., 1974, 21: 549–568.

    MATH  MathSciNet  Google Scholar 

  23. Lempel, A., Evens, S., Cederbaum, I., An algorithm for planarity testing of graphs, Theory of Graph (ed. Rosenstiehl, P.), New York: Gordon and Breach, 1967, 215–232.

    Google Scholar 

  24. Shih, W. K., Hsu, W. L., A simple test for planar graphs, Proceedings of the Third China-USA International Conference on Graph Theory, Beijing (eds. Alavi, Y., Lick, D. R., Liu Jinqiang), Singapore: World Scientific, 1994, 21–28.

    Google Scholar 

  25. Takefuji, Y., A near-optimum parallel planarizatio algorithm, Science, 1989, 245: 1221–1223.

    Article  Google Scholar 

  26. Xu Jin, Gao Lin, Zhang, J. Y., A new algorithm of planar graphs based on neural networks, Journal of Xidian University, 2001, (2): 1–5.

    Google Scholar 

  27. Wilson, G. V., Pawley, G. S., On stability of the traveling salesman problem algorithm of Hopfield and tank, Bio. Cybern., 1988, 58: 63–70.

    Article  MATH  MathSciNet  Google Scholar 

  28. Aiyer, S. V. B., Niranjan, M., Fallside, F., A theoretical investigation into the performance of the Hopfield model, IEEE Trans on Neural Network, 1990, 1(2): 204–215.

    Article  Google Scholar 

  29. Sun, S. Y., Zheng, J. L., A modified algorithm and theoretical analysis for Hopfield network solving TSP, Acta Electronica Sinica, 1995, 23(1): 73–78.

    Google Scholar 

  30. Zhang, Y. J., Ye, Z. X., Neural Networks with Applications—94' New Advance (in Chinese), Wuhan: The Huazhong University of Science and Technologicl Press, 1994.

    Google Scholar 

  31. Yu Daoheng, An improved neural network algorithm for TSP problem, Proceedings of the First Chinese Conference on Articifial Neural Networks (in Chinese), Beijing: China Neural Network Society, 1990, 116–121.

    Google Scholar 

  32. Jin Fan, Hu Fei, Chang Jianwu, A solution of C-TSP based on Hopfield Network, Proceedings of the First Chinese Conference on Articifial Neural Networks, Beijing (in Chinese), Beijing: China Neural Network Society, 1990, 122–129.

    Google Scholar 

  33. Fan Shemin, Ph. D. Thesis, Xi'an: Xi'an Jiaotong University, 1993.

  34. Sanchis, L., Multiple-way network partitioning, IEEE Trans. Computers, 1989, 38: 62–81.

    Article  MATH  Google Scholar 

  35. Fox, G. C., A review of automatic load balancing and decomposition methods for the Hypercube, Numerical Alorithms for Modern Parallel Computer Architectures (ed. Schultz, M.), New York: Springer-Verlag, 1988, 63–76.

    Google Scholar 

  36. Simon, H. D., Partitioning of unstructured problems for parallel processing, Computing Systems in Eng., 1991, 2: 135–148.

    Article  Google Scholar 

  37. Williams, R., Unification for spectral and intertial bisection, Technical Report, California Inst. of Technology, 1994, Available at: http://www.carcr.caltech.edu/roy/paper/

  38. Hendrickson, B., Leland, R., An improved spectral graph partitioning algorithm for mapping parallel computations, SIAM J. Scientific Computing, 1995, 16(2): 45–469.

    Article  MathSciNet  Google Scholar 

  39. Gilbert, J. R., Miller, G. L., Teng, S. H., Geometric mesh partitioning: Implementation and experiments, Proc. International First Parallel Processing Symp., (IPPS'95), 1995.

  40. Boppana, R. B., Eigenvalues and graph bisection: An average case analysis, Proc. 28th Symp. Foundations of Computer Science, 1987, 280–285.

  41. Bui, T. N., Chaudhuri, S., Leighton, F. T. et al., Graph bisection algorithms with good average case behavior, Combinatorica, 1987, 7(2): 171–191.

    Article  MathSciNet  Google Scholar 

  42. Bui, T. N., Peck, A., Partitioning planar graphs, SIAM J. Computing, 1992, 21(2): 203–215.

    Article  MATH  MathSciNet  Google Scholar 

  43. Garey, M., Johnson, D. S., Computers and Intractability: A Guide to the Theory of NP-Completeness, San Francisco: Freeman, 1979.

    MATH  Google Scholar 

  44. Bui, T. N., Jones, C., Finding good approximate vertex and edge partitions is NP-hard, Information Processing Letters, 1992, 42: 153–159.

    Article  MATH  MathSciNet  Google Scholar 

  45. Leighton, F. T., Rao, S., An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms, Proc. 29th Symp. Foundations of Computer Science, 1988, 422–431.

  46. Farhat, C., A simple and efficient automatic FEM domain decomposer, Computers and Structures, 1988, 28(5): 579–602.

    Article  Google Scholar 

  47. Farhat, C., Lanteri, S., Simon, H. D., TOP/DOMDEC—A software tool for mesh partitioning and parallel processing, J. Computing Systems in Eng., 1995, 6(1): 13–26.

    Article  Google Scholar 

  48. Laguna, M., Feo, T. A., Elrod, H. C., A greedy randomized adaptive search procedure for the two-partition problems, Operations Research, 1994, 42: 677–687.

    Article  MATH  Google Scholar 

  49. Battiti, R., Bertossi, A. A., Differential greedy for the 0–1 equicut problem, Network Design: Connectivity and Facilities Location (eds. Du, D. Z., Pardalos, P. M.), Princeton: Amer. Math. Soc., 1997, 3–22.

    Google Scholar 

  50. Johnson, D. S., Aragon, C. R., Mcgeoch, L. A. et al., Optimization by simulated annealing: An experimental evaluation; Part I, Graph Partitioning, Operations Research, 1989, 37: 865–892.

    MATH  Google Scholar 

  51. Kirkpatrick, S., Optimization by simulated annealing: Quantitative studies, J. Statistical Physics, 1994, 34: 975–986.

    Article  MathSciNet  Google Scholar 

  52. Kirkpatrick, S., Gelatt, C. D. Jr., Vecchi, M. P., Optimization by simulated annealing, Science, 1983 (May), 220 (4598): 671–680.

    Article  MathSciNet  Google Scholar 

  53. Rutebar, R., Simulated annealing algorithms: An overview, IEEE Circuit and Devices Magazine, 1989, 19–26.

  54. Sechen, C., Sangiovanni-Vincetelli, A., Timberwolf3.2: A new standard cell placement and global routing package, Proc. 23rd ACM/IEEE Design Automation Conf., 1986, 432–339.

  55. Sun, W., Sechen, C., Efficient and Effective placement for very large circuits, IEEE Trans. Computer-Aided Design, 1995, 14(3): 349–359.

    Article  Google Scholar 

  56. Battiti, R., Bertossi, A. A., Greedy, prohibition, and reactive heuristics for graph partitioning, IEEE Trans. on Computers, 1999, 48(4): 361–385.

    Article  Google Scholar 

  57. Bui, T. N., Moon, B. R., Genetic algorithm and grpah partitioning, IEEE Trans. Computers, 1996, 45(7): 841–855.

    Article  MATH  MathSciNet  Google Scholar 

  58. Kernighan, B., Lin, S., An efficient heuristic procedure for partitioning graphs, Bell Systems Technical J., 1970, 49(2): 291–307.

    Google Scholar 

  59. Fiduccia, C., Mattheyses, M., A linear time heuristics for improving network partitions, Proc. 19th ACM/IEEE Deisgn Automation Conf., Las Vegas, 1993, 175–181.

  60. Hendrickson, H., Leland, R., A multilevel algorithm for partitioning graphs, Technical Report SAND93-1301, SANDIA Nat'l Laboratory 1993.

  61. Diekmann, R., Monien, B., Preis, R., Using helpful sets to improve graph bisections, Interconnection Networks and Mapping and Scheduling Parallel Computations (eds. Hsu, D. F., Rosenberg, A. L., Sotteau, D.), Amer. Math. Soc., 1995, 57–73.

  62. Glover, F., Tabu Search-Part I, ORSA, J. Computing, 1989, 1(3): 190–260.

    MATH  Google Scholar 

  63. Rolland, E., Pirkul, H., Glover, F., A tabu search for graph partitioning, Annals of Operations Research, Metaheurisitcs in Combinatorial Optimization, 1996, 63: 46–53.

    Google Scholar 

  64. Dell'Amico, M., Maffioli, F., A new Tabu search approach to the 0–1 equicut problem, Meta-Heuristics 1995: The State of the Art, Dordrecht: Kluwer Academic, 1996, 361–377.

    Google Scholar 

  65. Pirkul, H., Rolland, E., New heuristic solution procedures for the uniform graph partitioning problem: Extensions and evaluation, Computers and Operations Research, 1994, 21(8): 895–907.

    Article  MATH  Google Scholar 

  66. Rolland, E., Pirkul, H., Heuristic solution procedures for the graph partitioning problem, Computer Science and Operations Research: New Developments in Their Interfaces (ed. Balci, O.), Oxford: Pergamon Press, 1992.

    Google Scholar 

  67. Bui, T. N., Jones, C., A heuristic for reducing fill in sparse matrix factorization, Proc. Sixth SIAM Conf. Parallel Processing for Scientific Computing, Philadelphia: SIAM, 1993, 445–452.

    Google Scholar 

  68. Garey, M. R., Johnson, D. S., Computers and Intractability: A Guide to the Theory of NP-completeness, San Francisco, New York: Freeman, 1979.

    MATH  Google Scholar 

  69. Orlova, G. I., Dorfman, Y. G., Finding the maximum cut in a graph, Eng. Cybern., 1972, 10: 502–506.

    MATH  MathSciNet  Google Scholar 

  70. Poljak, S., Turzik, D., Max-cut in circulant graphs, Discrete Mathemaics, 1992, 108: 379–392.

    Article  MATH  MathSciNet  Google Scholar 

  71. Hsu, C. P., Minimum-via topological routing, IEEE Trans. CAD/ICAS, 1983, (CAD) 2(4): 235–246.

    Google Scholar 

  72. Barahona, F., Grötschel, M., Jünger, M. et al., An application of combinatorial optimuzation to statistical physics and circuit layout design, Oper. Res., 1988, 36(3): 493–513.

    MATH  Google Scholar 

  73. Delorme, C., Poljak, S., Laplacian eigenvalues and the maximum cut problem, Math. Program., 1993, 62: 557–574.

    Article  MathSciNet  Google Scholar 

  74. Delorme, C., Poljak, S., The performance of an eigenvalue bound on the max-cut problem in some classes of graphs, Discrete Math., 1993, 111: 145–156.

    Article  MATH  MathSciNet  Google Scholar 

  75. Poljak, S., Rendl, F., Note and edge relaxations of the max-cut problem, Computing, 1994, 52: 123–137.

    Article  MATH  MathSciNet  Google Scholar 

  76. Korst, J. H. M., Aarts, E. H. L., Combinatorial optimization on a Boltzmann machine, J. Paral. Dist. Comput., 1989, 6: 331–357.

    Article  Google Scholar 

  77. Funabiki, N., Nishiawa, S., Tajima, S., A binary neural network approach for max cut problems, ICONIP'96-Hong Kong, Hong Kong: The Chinese Univ. Press, 1996, 631–635.

    Google Scholar 

  78. Battiti, R., Albert, B. A., Greedy, prohibition, and reactive heuristes fir graph partitioning, IEEE Trans. on Computers, 1999, 48(4): 361–385.

    Article  Google Scholar 

  79. Ramanujam, J., Sadayapan, P., Optimization by neural networks, Proc. IEEE Int'l Conf. on Neural Networks, II, 1988, 325–332.

    Article  Google Scholar 

  80. Foo, Y. S., Takefuji, Y., Stochastic neural networks for solving job-shop scheduling: Part I: problem presentation, Proc. IEEE Int'l Conf. on Neural Networks, II, San Diego, New York: IEEE, 1988, 283–290.

    Google Scholar 

  81. Chen Guoliang, Neurocomputing with applications in optimization, Journal of Computer Science and Development, 1992, (5): 1–21.

    Article  MATH  Google Scholar 

  82. Hopfield, J. J., Tank, D. W., Simple neural optimization nerworks: an A/D converter, signal decision circuit, and a linear programming circuit, IEEE Trans. on CAS, 1986, 33(5): 533–541.

    Google Scholar 

  83. Liou, C. Y., Chang, Q. M., Numerical soap film for the Steiner tree problem, ICONIP'96-Hong Kong, Hong Kong: The Chinese Univ. Press, 1996, 642–648.

    Google Scholar 

  84. Bruck, J., On the convergence properties of the Hopfield model, Proceedings of the IEEE, 1990, 78(10): 1579–1585.

    Article  Google Scholar 

  85. Goles, E., Antisymmetrical neural networks, Discrete Applied Mathematics, 1986, 13: 97–100.

    Article  MATH  MathSciNet  Google Scholar 

  86. Xu, J., Bao, Z., The stability of Antisymmetrical discrete Hopfield neural networks, Acta Electronica Sinica, 1999, 27(1): 103–107.

    Google Scholar 

  87. Xu Jin, Bao, Z., The linear separability of a Boolean function (I)—the fundational ofn-dimensional hypercubes, Journal of Electronics, 1996, 18: 6–13.

    Google Scholar 

  88. Xu Jin, Bao, Z., Linear separability of Boolean function (II), Journal of Electronics, 1996, 18: 14–21

    Google Scholar 

  89. Goles, E., Fogelman, F., Pellegrin, D., Decreasing energy functions as a tool for studying threshold networks, Discrete Applied Mathematics, 1985, 12: 261–277.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Control Science and Engineering, Huazhong University of Science and Technology, 430074, Wuhan, China

    Jin Xu

  2. Electronic Engineering Research Institute, University of Electronic Science and Technology, 710071, Xi'an, China

    Zheng Bao

Authors
  1. Jin Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zheng Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin Xu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, J., Bao, Z. Neural networks and graph theory. Sci China Ser F 45, 1–24 (2002). https://doi.org/10.1360/02yf9001

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1360/02yf9001

Keywords

Search

Navigation