On Graph Kernels: Hardness Results and Efficient Alternatives

  • Thomas Gärtner
  • Peter Flach
  • Stefan Wrobel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2777)


As most ‘real-world’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. An interesting and important challenge is thus to investigate kernels on instances that are represented by graphs. So far, only very specific graphs such as trees and strings have been considered.

This paper investigates kernels on labeled directed graphs with general structure. It is shown that computing a strictly positive definite graph kernel is at least as hard as solving the graph isomorphism problem. It is also shown that computing an inner product in a feature space indexed by all possible graphs, where each feature counts the number of subgraphs isomorphic to that graph, is NP-hard. On the other hand, inner products in an alternative feature space, based on walks in the graph, can be computed in polynomial time. Such kernels are defined in this paper.


Feature Space Adjacency Matrix Product Graph Hamiltonian Path Transition Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Boser, B.E., Guyon, I.M., Vapnik, V.N.: A training algorithm for optimal margin classifiers. In: Haussler, D. (ed.) Proceedings of the 5th Annual ACM Workshop on Computational Learning Theory, Pittsburgh, PA, July 1992, pp. 144–152. ACM Press, New York (1992)CrossRefGoogle Scholar
  2. 2.
    Collins, M., Duffy, N.: Convolution kernels for natural language. In: Dietterich, T.G., Becker, S., Ghahramani, Z. (eds.) Advances in Neural Information Processing Systems, Cambridge, MA, vol. 14. MIT Press, Cambridge (2002)Google Scholar
  3. 3.
    Diestel, R.: Graph Theory. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    Gärtner, T.: Exponential and geometric kernels for graphs. In: NIPS Workshop on Unreal Data: Principles of Modeling Nonvectorial Data (2002)Google Scholar
  5. 5.
    Gärtner, T.: Kernel-based multi-relational data mining. In: SIGKDD Explorations (2003) (to appear)Google Scholar
  6. 6.
    Gärtner, T., Driessens, K., Ramon, J.: Graph kernels and gaussian processes for relational reinforcement learning. In: Proceedings of the 13th International Conference on Inductive Logic Programming (2003) (submitted)Google Scholar
  7. 7.
    Gray, R.: Probability, Random Processes, and Ergodic Properties. Springer, Heidelberg (1987)Google Scholar
  8. 8.
    Imrich, W., Klavžar, S.: Product Graphs: Structure and Recognition. John Wiley, Chichester (2000)zbMATHGoogle Scholar
  9. 9.
    Kashima, H., Inokuchi, A.: Kernels for graph classification. In: ICDM Workshop on Active Mining (2002)Google Scholar
  10. 10.
    Kashima, H., Tsuda, K., Inokuchi, A.: Marginalized kernels between labeled graphs. In: Proceedings of the 20th International Conference on Machine Learning (2003) (to appear)Google Scholar
  11. 11.
    Köbler, J., Schöning, U., Töran, J.: The Graph Isomorphism Problem: Its Structural Complexity. Progress in Theoretical Computer Science. Birkhäuser, Basel (1993)Google Scholar
  12. 12.
    Kondor, R.I., Lafferty, J.: Diffusion kernels on graphs and other discrete input spaces. In: Sammut, C., Hoffmann, A. (eds.) Proceedings of the 19th International Conference on Machine Learning, pp. 315–322. Morgan Kaufmann, San Francisco (2002)Google Scholar
  13. 13.
    Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  14. 14.
    Lodhi, H., Saunders, C., Shawe-Taylor, J., Cristianini, N., Watkins, C.: Text lassification using string kernels. Journal of Machine Learning Research 2, 419–444 (2002)zbMATHCrossRefGoogle Scholar
  15. 15.
    Schölkopf, B., Smola, A.J.: Learning with Kernels. The MIT Press, Cambridge (2002)Google Scholar
  16. 16.
    Skena, S.: The Algorithm Design Manual. Springer, Heidelberg (1997)Google Scholar
  17. 17.
    Watkins, C.: Kernels from matching operations. Technical report, Department of Computer Science, Royal Holloway, University of London (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Thomas Gärtner
    • 1
    • 2
    • 3
  • Peter Flach
    • 3
  • Stefan Wrobel
    • 1
    • 2
  1. 1.Fraunhofer Institut Autonome Intelligente SystemeGermany
  2. 2.Department of Computer Science IIIUniversity of BonnGermany
  3. 3.Department of Computer ScienceUniversity of BristolUK

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