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A polynomial-time maximum common subgraph algorithm for outerplanar graphs and its application to chemoinformatics

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Abstract

Metrics for structured data have received an increasing interest in the machine learning community. Graphs provide a natural representation for structured data, but a lot of operations on graphs are computationally intractable. In this article, we present a polynomial-time algorithm that computes a maximum common subgraph of two outerplanar graphs. The algorithm makes use of the block-and-bridge preserving subgraph isomorphism, which has significant efficiency benefits and is also motivated from a chemical perspective. We focus on the application of learning structure-activity relationships, where the task is to predict the chemical activity of molecules. We show how the algorithm can be used to construct a metric for structured data and we evaluate this metric and more generally also the block-and-bridge preserving matching operator on 60 molecular datasets, obtaining state-of-the-art results in terms of predictive performance and efficiency.

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References

  1. Akutsu, T.: A polynomial time algorithm for finding a largest common subgraph of almost trees of bounded degree. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E76-A, 1488–1493 (1993)

    Google Scholar 

  2. Bringmann, B., Zimmermann, A., De Raedt, L., Nijssen, S.: Don’t be afraid of simpler patterns. In: Proceedings of the 10th European Conference on Principles and Practice of Knowledge Discovery in Databases, pp. 55–66 (2006)

  3. Bunke, H., Shearer, K.: A graph distance metric based on the maximal common subgraph. Pattern Recogn. Lett. 19(3–4), 255–259 (1998)

    Article  MATH  Google Scholar 

  4. Cao, Y., Jiang, T., Girke, T.: A maximum common substructure-based algorithm for searching and predicting drug-like compounds. Bioinformatics 24(13), i366–i374 (2008)

    Article  Google Scholar 

  5. Ceroni, A., Costa, F., Frasconi, P.: Classification of small molecules by two- and three-dimensional decomposition kernels. Bioinformatics 23(16), 2038–2045 (2007)

    Article  Google Scholar 

  6. Chaoji, V., Al Hasan, M., Salem, S., Besson, J., Zaki, M.J.: Origami: A novel and effective approach for mining representative orthogonal graph patterns. Stat. Anal. Data Min. 1(2), 67–84 (2008)

    Article  MathSciNet  Google Scholar 

  7. Chi, Y., Muntz, R.R., Nijssen, S., Kok, J.N.: Frequent subtree mining—an overview. Fundam. Inform. 66(1–2), 161–198 (2005)

    MATH  MathSciNet  Google Scholar 

  8. Conte, D., Foggia, P., Sansone, C., Vento, M.: Thirty years of graph matching in pattern recognition. Int. J. Pattern Recogn. Artif. Intell. 18(3), 265–298 (2004)

    Article  Google Scholar 

  9. De Raedt, L.: Logical and Relational Learning. Springer (2008)

  10. De Raedt, L., Ramon, J.: Deriving distance metrics from generality relations. Pattern Recogn. Lett. 30(3), 187–191 (2009)

    Article  Google Scholar 

  11. Demšar, J.: Statistical comparisons of classifiers over multiple data sets. J. Mach. Learn. Res. 7, 1–30 (2006)

    MATH  MathSciNet  Google Scholar 

  12. Deshpande, M., Kuramochi, M., Wale, N., Karypis, G.: Frequent substructure-based approaches for classifying chemical compounds. IEEE Trans. Knowl. Data Eng. 17(8), 1036–1050 (2005)

    Article  Google Scholar 

  13. Diestel, R.: Graph Theory. Springer-Verlag (2000)

  14. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman and Co. (1979)

  15. Gärtner, T.: Kernels for Structured Data. World Scientific (2008)

  16. Hansch, C., Maolney, P.P., Fujita, T., Muir, R.M.: Correlation of biological activity of phenoxyacetic acids with hammett substituent constants and partition coefficients. Nature 194, 178–180 (1962)

    Article  Google Scholar 

  17. He, H., Singh, A.K.: Graphrank: statistical modeling and mining of significant subgraphs in the feature space. In: ICDM ’06: Proceedings of the 6th International Conference on Data Mining, pp. 885–890. IEEE Computer Society, Washington, DC (2006)

    Google Scholar 

  18. Helma, C., Kramer S., De Raedt, L: Data mining and machine learning techniques for the identification of mutagenicity inducing substructures and structure activity relationships of noncongeneric compounds. J. Chem. Inf. Model. 44(4), 1402–141 (2004)

    Article  Google Scholar 

  19. Hopcroft, J.E., Karp, R.M.: A n 5/2 algorithm for maximum matching in bipartite graphs. SIAM J. Comput. 2, 225–231 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  20. Horváth, T., Gärtner, T., Wrobel, S.: Cyclic pattern kernels for predictive graph mining. In: KDD ’04: Proceedings of the 10th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 158–167 (2004)

  21. Horváth, T., Ramon, J., Wrobel, S.: Frequent subgraph mining in outerplanar graphs. In: KDD ’06: Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 197–206. Philadelphia, PA (2006)

  22. Horváth, T., Ramon, J., Wrobel, S.: Frequent subgraph mining in outerplanar graphs. Data Min. Knowl. Discov. 21(3), 472–508 (2010)

    Article  MathSciNet  Google Scholar 

  23. Joachims, T.: Learning to Classify Text using Support Vector Machines: Methods, Theory, and Algorithms. Springer (2002)

  24. Johnson, M.A., Maggiora, G.M.: Concepts and Applications of Molecular Similarity. John Wiley (1990)

  25. Karunaratne, T., Boström, H.: Learning to classify structured data by graph propositionalization. In: Proceedings of the 2nd IASTED International Conference on Computational Intelligence, pp. 393–398 (2006)

  26. King, R.D., Muggleton, S., Srinivasan, A., Sternberg, M.J.E.: Structure-activity relationships derived by machine learning: the use of atoms and their bond connectivities to predict mutagenicity by inductive logic programming. Proc. Natl. Acad. Sci. 93, 438–442 (1996)

    Article  Google Scholar 

  27. Koch, I.: Enumerating all connected maximal common subgraphs in two graphs. Theor. Comput. Sci. 250(1–2), 1–30 (2001)

    Article  MATH  Google Scholar 

  28. Kramer, S., De Raedt, L., Helma, C.: Molecular feature mining in HIV data. In: Proceedings of the 7th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD-01), pp. 136–143. ACM Press (2001)

  29. Kramer, S., Lavrač, N., Flach, P.: Propositionalization approaches to relational data mining. In: Džeroski, S., Lavrač, N. (eds.) Relational Data Mining, pp. 262–291. Springer-Verlag (2001)

  30. Lingas, A.: Subgraph isomorphism for biconnected outerplanar graphs in cubic time. Theor. Comput. Sci. 63, 295–302 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  31. Maunz, A., Helma, C., Kramer, S.: Large-scale graph mining using backbone refinement classes. In: KDD ’09: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 617–626. ACM, New York, NY (2009)

    Chapter  Google Scholar 

  32. McGregor, J.J.: Backtrack search algorithms and the maximal common subgraph problem. Softw. Pract. Exp. 12, 23–34 (1982)

    Article  MATH  Google Scholar 

  33. Mitchell, S.L.: Linear algorithms to recognize outerplanar and maximal outerplanar graphs. Inf. Process. Lett. 9(5), 229–232 (1979)

    Article  MATH  Google Scholar 

  34. Munkres, J.: Algorithms for the assignment and transportation problems. J. Soc. Ind. Appl. Math. 5(1), 32–38 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  35. Nijssen, S., Kok, J.N.: A quickstart in frequent structure mining can make a difference. In: Proceedings of the 10th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), pp. 647–652 (2004)

  36. Raymond, J., Gardiner, E., Willett, P.: Rascal: calculation of graph similarity using maximum common edge subgraphs. Comput. J. 45, 631–644 (2002)

    Article  MATH  Google Scholar 

  37. Raymond, J., Willett, P.: Effectiveness of graph-based and fingerprint-based similarity measures for virtual screening of 2D chemical structure databases. J. Comput. Aided Mol. Des. 16, 59–71 (2002)

    Article  Google Scholar 

  38. Raymond, J., Willett, P.: Maximum common subgraph isomorphism algorithms for the matching of chemical structures. J. Comput. Aided Mol. Des. 16, 521–533 (2002)

    Article  Google Scholar 

  39. Schietgat, L., Ramon, J., Bruynooghe, M., Blockeel, H.: An efficiently computable graph-based metric for the classification of small molecules. In: Proceedings of the 11th International Conference on Discovery Science, vol. 5255 of Lecture Notes in Artificial Intelligence, pp. 197–209 (2008)

  40. Schietgat, L., Costa, F., Ramon, J., De Raedt, L.: Effective feature construction by maximum common subgraph sampling. Mach. Learn. 83(2), 137–161 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  41. Shamir, R., Tsur, D.: Faster subtree isomorphism. J. Algorithms 33(2), 267–280 (1992)

    Article  MathSciNet  Google Scholar 

  42. Shearer, K., Bunke, H., Venkatesh, S.: Video indexing and similarity retrieval by largest common subgraph detection using decision trees. Pattern Recogn. 34(5), 1075–1091 (2001)

    Article  MATH  Google Scholar 

  43. Shervashidze, N., Borgwardt, K.: Fast subtree kernels on graphs. In: Bengio, Y., Schuurmans, D., Lafferty, J., Williams, C.K.I., Culotta, A. (eds.) Advances in Neural Information Processing Systems, vol. 22, pp. 1660–1668 (2009)

  44. Swamidass, S.J., Chen, J., Bruand, J., Phung, P., Ralaivola, L., Baldi, P.: Kernels for small molecules and the prediction of mutagenicity, toxicity and anti-cancer activity. Bioinformatics 21, i359–i368 (2005)

    Article  Google Scholar 

  45. Syslo, M.: The subgraph isomorphism problem for outerplanar graphs. Theor. Comp. Sci. 17(1), 91–97 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  46. Wale, N., Watson, I.A., Karypis, G.: Comparison of descriptor spaces for chemical compound retrieval and classification. Knowl. Inf. Syst. 14, 347–375 (2008)

    Article  Google Scholar 

  47. Wilcoxon, F.: Individual comparisons by ranking methods. Biometrics 1, 80–83 (1945)

    Article  Google Scholar 

  48. Willett, P.: Similarity-based virtual screening using 2D fingerprints. Drug Discov. Today 11(23/24), 1046–1051 (2006)

    Article  Google Scholar 

  49. Yan, X., Han, J.: gSpan: graph-based substructure pattern mining. In: Proceedings of the 2002 IEEE International Conference on Data Mining (ICDM 2002), pp. 721–724. IEEE Computer Society (2002)

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Authors and Affiliations

  1. Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001, Leuven, Belgium

    Leander Schietgat, Jan Ramon & Maurice Bruynooghe

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  1. Leander Schietgat
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  2. Jan Ramon
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  3. Maurice Bruynooghe
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Correspondence to Leander Schietgat.

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Schietgat, L., Ramon, J. & Bruynooghe, M. A polynomial-time maximum common subgraph algorithm for outerplanar graphs and its application to chemoinformatics. Ann Math Artif Intell 69, 343–376 (2013). https://doi.org/10.1007/s10472-013-9335-0

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