Continuous-based Heuristics for Graph and Tree Isomorphisms, with Application to Computer Vision

  • Marcello Pelillo
  • Kaleem Siddiqi
  • Steven W. Zucker
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 42)


We present a new (continuous) quadratic programming approach for graph- and tree-isomorphism problems which is based on an equivalent maximum clique formulation. The approach is centered around a fundamental result proved by Motzkin and Straus in the mid-1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program. The attractive feature of this formulation is that a clear one-to-one correspondence exists between the solutions of the quadratic programs and those in the original, combinatorial problems. To approximately solve the program we use the so-called “replicator” equations, a class of straightforward continuous- and discrete-time dynamical systems developed in various branches of theoretical biology. We show how, despite their inherent inability to escape from local solutions, they nevertheless provide experimental results which are competitive with those obtained using more sophisticated mean-field annealing heuristics. Application of this approach to shape matching problems arising in computer vision and pattern recognition are also presented.


Maximum clique quadratic programming replicator dynamics shape recognition. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardness of approximation problems. In Proc. 33rd Ann. Symp. Found. Comput. Sci., pages 14–23. Pittsburgh, PA, 1992.CrossRefGoogle Scholar
  2. [2]
    L. Babai, P. Erdös, and S. M. Selkow. Random graph isomorphism. SIAM J. Comput., 9 (3): 628–635, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    H. G. Barrow and R. M. Burstall. Subgraph isomorphism, matching relational structures, and maximal cliques. Inform. Process. Lett., 4 (4): 83–84, 1976.zbMATHCrossRefGoogle Scholar
  4. [4]
    L. E. Baum and J. A. Eagon. An inequality with applications to statistical estimation for probabilistic functions of markov processes and to a model for ecology. Bull. Amer. Math. Soc., 73: 360–363, 1967.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    I. M. Bomze. Evolution towards the maximum clique. J. Global Optim., 10: 143164, 1997.Google Scholar
  6. [6]
    I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, Boston, MA, 1999.Google Scholar
  7. [7]
    I. M. Bomze, M. Budinich, M. Pelillo, and C. Rossi. Annealed replication: A new heuristic for the maximum clique problem. Discr. Appl. Math.,1999. to appear.Google Scholar
  8. [8]
    I. M. Bomze, M. Pelillo, and R. Giacomini Evolutionary approach to the maximum clique problem: Empirical evidence on a larger scale. In I. M. Bomze, T. Csendes, R. Horst, and P. M. Pardalos, editors, Developments in Global Optimization, pages 95–108. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.CrossRefGoogle Scholar
  9. [9]
    R. B. Boppana, J. Hastad, and S. Zachos. Does co-NP have short interactive proofs? Inform. Process. Lett., 25: 127–132, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    R. Brockett and P. Maragos. Evolution equations for continuous-scale morphology. In Proceedings of the IEEE Conference on Acoustics, Speech and Signal Processing, San Francisco, CA, March 1992.Google Scholar
  11. [11]
    R. Durbin and D. Willshaw. An analog approach to the travelling salesman problem using an elastic net method. Nature, 326: 689–691, 1987.CrossRefGoogle Scholar
  12. [12]
    Y. Fu and P. W. Anderson. Application of statistical mechanics to NP-complete problems in combinatorial optimization. J. Phys. A, 19: 1605–1620, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco, CA, 1979.Google Scholar
  14. [14]
    L. E. Gibbons, D. W. Hearn, and P. M. Pardalos. A continuous based heuristic for the maximum clique problem. In D. S. Johnson and M. Trick, editors, Cliques, Coloring, and Satisfiability—Second DIMACS Implementation Challenge, pages 103–124. American Mathematical Society, 1996.Google Scholar
  15. [15]
    L. E. Gibbons, D. W. Hearn, P. M. Pardalos, and M. V. Ramana. Continuous characterizations of the maximum clique problem. Math. Oper. Res., 22: 754–768, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    S. Gold and A. Rangarajan. A graduated assignment algorithm for graph matching. IEEE Trans. Pattern Anal. Machine Intell., 18 (4): 377–388, 1996.CrossRefGoogle Scholar
  17. [17]
    M. Grötschel, L. Lovâsz, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin, 1988.zbMATHCrossRefGoogle Scholar
  18. [18]
    F. Harary. Graph Theory. Addison-Wesley, Reading, MA, 1969.Google Scholar
  19. [19]
    J. Hastad. Clique is hard to approximate within n1—E. In Proc. 37th Ann. Symp. Found. Comput. Sci., pages 627–636, 1996.Google Scholar
  20. [20]
    J. Hofbauer. Imitation dynamics for games. Collegium Budapest, preprint, 1995.Google Scholar
  21. [21]
    J. Hofbauer and K. Sigmund. The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge, UK, 1988.zbMATHGoogle Scholar
  22. [22]
    J. J. Hopfield and D. W. Tank. Neural computation of decisions in optimization problems. Biol. Cybern., 52: 141–152, 1985.MathSciNetzbMATHGoogle Scholar
  23. [23]
    A. Jagota. Approximating maximum clique with a Hopfield network. IEEE Trans. Neural Networks, 6: 724–735, 1995.CrossRefGoogle Scholar
  24. [24]
    D. S. Johnson. The NP-completeness column: An ongoing guide. J. Algorithms, 9: 426–444, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    B. B. Kimia, A. Tannenbaum, and S. W. Zucker. Shape, shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space. Int. J. Comp. Vision, 15: 189–224, 1995.CrossRefGoogle Scholar
  26. [26]
    J. J. Kosowsky and A. L. Yuille. The invisible hand algorithm: Solving the assignment problem with statistical physics. Neural Networks, 7: 477–490, 1994.zbMATHCrossRefGoogle Scholar
  27. [27]
    D. Kozen. A clique problem equivalent to graph isomorphism. SIGACT News, pages 50–52, Summer 1978.Google Scholar
  28. [28]
    P. D. Lax. Shock waves and entropy. In E. H. Zarantonello, editor, Contributions to Nonlinear Functional Analysis, pages 603–634, New York, 1971. Acad. Press.Google Scholar
  29. [29]
    J. T. Li, K. Zhang, K. Jeong, and D. Shasha. A system for approximate tree matching. IEEE Trans. Knowledge Data Eng., 6: 559–571, 1994.CrossRefGoogle Scholar
  30. [30]
    V. Losert and E. Akin. Dynamics of games and genes: Discrete versus continuous time. J. Math. Biol., 17: 241–251, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    S. Y. Lu. A tree-matching algorithm based on node splitting and merging. IEEE Trans. Pattern Anal. Machine Intell., 6: 249–256, 1984.zbMATHCrossRefGoogle Scholar
  32. [32]
    D. Marr and K. H. Nishihara. Representation and recognition of the spatial organization of three-dimensional shapes. Proc. R. Soc. Lond. B, 200: 269–294, 1978.CrossRefGoogle Scholar
  33. [33]
    D. W. Matula. An algorithm for subtree identification. Siam Rev., 10: 273–274, 1968.Google Scholar
  34. [34]
    T. S. Motzkin and E. G. Straus. Maxima for graphs and a new proof of a theorem of Turd’’’. Canad. J. Math., 17: 533–540, 1965.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    M. Neff, R. Byrd, and O. Rizk. Creating and querying hierarchical lexical databases. In Proc. 2nd Conf. Appl. Natural Language Process., pages 84–93, 1988.CrossRefGoogle Scholar
  36. [36]
    M. Ohlsson, C. Peterson, and B. Söderberg. Neural networks for optimization problems with inequality constraints: The knapsack problem. Neural Computation, 5: 331–339, 1993.CrossRefGoogle Scholar
  37. [37]
    E. M. Palmer. Graphical Evolution: An Introduction to the Theory of Random Graphs. John Wiley & Sons, New York, 1985.zbMATHGoogle Scholar
  38. [38]
    P. M. Pardalos. Continuous approaches to discrete optimization problems. In G. D. Pillo and F. Giannessi, editors, Nonlinear Optimization and Applications, pages 313–328. Plenum Press, 1996.Google Scholar
  39. [39]
    P. M. Pardalos and A. T. Phillips. A global optimization approach for solving the maximum clique problem. Int. J. Comput. Math., 33: 209–216, 1990.zbMATHCrossRefGoogle Scholar
  40. [40]
    M. Pelillo. Relaxation labeling networks for the maximum clique problem. J. Artif. Neural Networks, 2: 313–328, 1995.Google Scholar
  41. [41]
    M. Pelillo. Replicator equations, maximal cliques, and graph isomorphism. Neural Computation, 11 (8): 2023–2045, 1999.CrossRefGoogle Scholar
  42. [42]
    M. Pelillo and A. Jagota. Feasible and infeasible maxima in a quadratic program for maximum clique. J. Artif. Neural Networks, 2: 411–420, 1995.Google Scholar
  43. [43]
    M. Pelillo, K. Siddiqi, and S. W. Zucker. Attributed tree matching and maximum weight cliques. In Proc. ICIAP’99–10th Int. Conf. on Image Analysis and Processing. IEEE Computer Society Press, 1999.Google Scholar
  44. [44]
    A. Rangarajan, S. Gold, and E. Mjolsness. A novel optimizing network architecture with applications. Neural Computation, 8: 1041–1060, 1996.CrossRefGoogle Scholar
  45. [45]
    A. Rangarajan and E. Mjolsness. A lagrangian relaxation network for graph matching. IEEE Trans. Neural Networks, 7 (6): 1365–1381, 1996.CrossRefGoogle Scholar
  46. [46]
    S. W. Reyner. An analysis of a good algorithm for the subtree problem. SIAM J. Comput., 6: 730–732, 1977.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    H. Rom and G. Medioni. Hierarchical decomposition and axial shape description. IEEE Trans. Pattern Anal. Machine Intell., 15 (10): 973–981, 1993.CrossRefGoogle Scholar
  48. [48]
    H. Samet. Design and Analysis of Spatial Data Structures. Addison-Wesley, Reading, MA, 1990.Google Scholar
  49. [49]
    U. Schöning. Graph isomorphism is in the low hierarchy. J. Comput. Syst. Sci., 37: 312–323, 1988.zbMATHCrossRefGoogle Scholar
  50. [50]
    B. A. Shapiro and K. Zhang. Comparing multiple RNA secondary structures using tree comparisons. Comput. Appl. Biosci., 6: 309–318, 1990.Google Scholar
  51. [51]
    D. Shasha, J. T. L. Wang, K. Zhang, and F. Y. Shih. Exact and approximate algorithms for unordered tree matching. IEEE Trans. Syst. Man Cybern., 24: 668678, 1994.Google Scholar
  52. [52]
    K. Siddiqi, A. Shokoufandeh, S. J. Dickinson, and S. W. Zucker. Shock graphs and shape matching. Int. J. Comp. Vision, to appear, 1999.Google Scholar
  53. [53]
    P. D. Simié. Constrained nets for graph matching and other quadratic assignment problems. Neural Computation, 3: 268–281, 1991.CrossRefGoogle Scholar
  54. [54]
    J. W. Weibull. Evolutionary Game Theory. MIT Press, Cambridge, MA, 1995.zbMATHGoogle Scholar
  55. [55]
    H. S. Wilf. Spectral bounds for the clique and independence numbers of graphs. J. Combin. Theory, Ser. B, 40: 113–117, 1986.MathSciNetzbMATHGoogle Scholar
  56. [56]
    S. Zhu and A. L. Yuille. FORMS: A flexible object recognition and modeling system. Int. J. Comp. Vision, 20 (3): 187–212, 1996.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Marcello Pelillo
    • 1
  • Kaleem Siddiqi
    • 2
  • Steven W. Zucker
    • 3
  1. 1.Dipartimento di InformaticaUniversità Ca’ Foscari di VeneziaItaly
  2. 2.Center for Intelligent MachinesMcGill UniversityCanada
  3. 3.Center for Computational Vision and ControlYale UniversityUSA

Personalised recommendations