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. Tinhofer (Disc. App. Math., 1991), Ramana, Scheinerman, and Ullman (Disc. Math., 1994), and Godsil (Lin. Alg. and its App., 1997) established a tight correspondence between colour refinement and fractional automorphisms of a graph. We introduce versions of colour refinement for weighted directed graphs, for matrices, and for linear programs and extend existing quasilinear algorithms for computing the colour classes. Then we generalise the correspondence between colour refinement and fractional automoprphisms, giving a new proof that is much simpler than the known proofs even in the setting of unweighted undirected graphs. We apply these results to reduce the dimensions linear programs. Specifically, we show that any given linear program L can efficiently be transformed into a (potentially) smaller linear program L' whose number of variables and constraints is the number of colour classes of the colour refinement algorithm. (When applied to a linear program, colour refinement yields partitions both of the constraints and of the variables.) The transformation is such that we can easily (by a linear mapping) transform 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. A precusor of the method proposed here has been applied successfully by the second and third author (jointly with Ahmadi) to MAP inference problems in machine learning (AISTATS 2012).
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