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)

Abstract

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.

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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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