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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References
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)
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)
Bödi, R., Grundhöfer, T., Herr, K.: Symmetries of linear programs. Note di Matematica 30(1), 129–132 (2010)
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)
Cardon, A., Crochemore, M.: Partitioning a graph in O(|A|log2|V|). Theoretical Computer Science 19(1), 85–98 (1982)
Godsil, C.D.: Compact graphs and equitable partitions. Linear Algebra and its Applications 255, 259–266 (1997)
Grohe, M., Kersting, K., Mladenov, M., Selman, E.: Dimension reduction via colour refinement. ArXiv, 1307.5697 (2014) (full version of this paper)
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)
Kersting, K., Ahmadi, B., Natarajan, S.: Counting Belief Propagation. In: Proc. of the 25th Conf. on Uncertainty in Artificial Intelligence, UAI 2009 (2009)
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)
Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM Journal on Computing 16(6), 973–989 (1987)
Ramana, M.V., Scheinerman, E.R., Ullman, D.: Fractional isomorphism of graphs. Discrete Mathematics 132, 247–265 (1994)
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)
Tinhofer, G.: A note on compact graphs. Discrete Applied Mathematics 30, 253–264 (1991)
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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
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