Skip to main content
View expanded cover

European Symposium on Algorithms

ESA 2014: Algorithms - ESA 2014 pp 505–516Cite as

Dimension Reduction via Colour Refinement

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8737)

Abstract

Colour refinement is a basic algorithmic routine for graph isomorphism testing, appearing as a subroutine in almost all practical isomorphism solvers. It partitions the vertices of a graph into “colour classes” in such a way that all vertices in the same colour class have the same number of neighbours in every colour class. There is a tight correspondence between colour refinement and fractional isomorphisms of graphs, which are solutions to the LP relaxation of a natural ILP formulation of graph isomorphism.

We introduce a version of colour refinement for matrices and extend existing quasilinear algorithms for computing the colour classes. Then we generalise the correspondence between colour refinement and fractional automorphisms and develop a theory of fractional automorphisms and isomorphisms of matrices.

We apply our results to reduce the dimensions of systems of linear equations and linear programs. Specifically, we show that any given LP L can efficiently be transformed into a (potentially) smaller LP L′ whose number of variables and constraints is the number of colour classes of the colour refinement algorithm, applied to a matrix associated with the LP. The transformation is such that we can easily (by a linear mapping) map both feasible and optimal solutions back and forth between the two LPs. We demonstrate empirically that colour refinement can indeed greatly reduce the cost of solving linear programs.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-662-44777-2_42
  • Chapter length: 12 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   89.00
Price excludes VAT (USA)
  • ISBN: 978-3-662-44777-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   119.99
Price excludes VAT (USA)
Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahmadi, B., Kersting, K., Mladenov, M., Natarajan, S.: Exploiting symmetries for scaling loopy belief propagation and relational training. Machine Learning Journal 92, 91–132 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Berkholz, C., Bonsma, P., Grohe, M.: Tight lower and upper bounds for the complexity of canonical colour refinement. In: Proceedings of the 21st Annual European Symposium on Algorithms (2013) (to appear)

    Google Scholar 

  3. Bödi, R., Grundhöfer, T., Herr, K.: Symmetries of linear programs. Note di Matematica 30(1), 129–132 (2010)

    MathSciNet  Google Scholar 

  4. Bui, H.H., Huynh, T.N., Riedel, S.: Automorphism groups of graphical models and lifted variational inference. In: Proc. of the 29th Conference on Uncertainty in Artificial Intelligence, UAI-2013 (2013)

    Google Scholar 

  5. Cardon, A., Crochemore, M.: Partitioning a graph in O(|A|log2|V|). Theoretical Computer Science 19(1), 85–98 (1982)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. Godsil, C.D.: Compact graphs and equitable partitions. Linear Algebra and its Applications 255, 259–266 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Grohe, M., Kersting, K., Mladenov, M., Selman, E.: Dimension reduction via colour refinement. ArXiv, 1307.5697 (2014) (full version of this paper)

    Google Scholar 

  8. Hopcroft, J.E.: An n log n algorithm for minimizing states in a finite automaton. In: Kohavi, Z., Paz, A. (eds.) Theory of Machines and Computations, pp. 189–196. Academic Press (1971)

    Google Scholar 

  9. Kersting, K., Ahmadi, B., Natarajan, S.: Counting Belief Propagation. In: Proc. of the 25th Conf. on Uncertainty in Artificial Intelligence, UAI 2009 (2009)

    Google Scholar 

  10. Mladenov, M., Ahmadi, B., Kersting, K.: Lifted linear programming. In: 15th Int. Conf. on Artificial Intelligence and Statistics (AISTATS 2012). JMLR: W&CP 22, vol. 22, pp. 788–797 (2012)

    Google Scholar 

  11. Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM Journal on Computing 16(6), 973–989 (1987)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. Ramana, M.V., Scheinerman, E.R., Ullman, D.: Fractional isomorphism of graphs. Discrete Mathematics 132, 247–265 (1994)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Singla, P., Domingos, P.: Lifted First-Order Belief Propagation. In: Proc. of the 23rd AAAI Conf. on Artificial Intelligence (AAAI 2008), Chicago, IL, USA, July 13-17, pp. 1094–1099. AAAI Press, Menlo Park (2008)

    Google Scholar 

  14. Tinhofer, G.: A note on compact graphs. Discrete Applied Mathematics 30, 253–264 (1991)

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. RWTH Aachen University, Germany

    Martin Grohe & Erkal Selman

  2. TU Dortmund University, Germany

    Kristian Kersting & Martin Mladenov

Authors
  1. Martin Grohe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kristian Kersting
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martin Mladenov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Erkal Selman
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, USA

    Andreas S. Schulz

  2. Karlsruhe Institute of Technology (KIT), Kaiserstruhe 12, 76131, Karlsruhe, Germany

    Dorothea Wagner

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Grohe, M., Kersting, K., Mladenov, M., Selman, E. (2014). Dimension Reduction via Colour Refinement. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_42

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-44777-2_42

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44776-5

  • Online ISBN: 978-3-662-44777-2

  • eBook Packages: Computer ScienceComputer Science (R0)