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European Symposium on Algorithms

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

Dimension Reduction via Colour Refinement

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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.

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  • DOI: 10.1007/978-3-662-44777-2_42
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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
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  2. Kristian Kersting
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  3. Martin Mladenov
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  4. Erkal Selman
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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

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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

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  • 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

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