Dimension Reduction via Colour Refinement

Grohe, Martin; Kersting, Kristian; Mladenov, Martin; Selman, Erkal

doi:10.48550/arxiv.1307.5697

Cited by 1 publication

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“…This converges in at most a linear number of iterations since in each step we either create a new color or we stop; overall, however, there can only be at most n colors. Finally, CGCR fulfills conditions (1.1) and (1.2) of (Grohe et al 2013) and hence converges to the coarsest equitable partition.…”

Section: Discussionmentioning

confidence: 82%

“…However, solving Ramana et al's LP using off-the-shelf LP solvers does not mimic CR -they differ in the solution as well as the path to the solution -and the graph theory view already led to quasi-linear O((m + n) log n) algorithms for finding FGAs of (weighted) graphs with n vertices and m edges due to asynchronous color updates, see e.g. (Berkholz, Bonsma, and Grohe 2013;Grohe et al 2013) for more details. So, is there a fundamental gap between the combinatorial and continuous views on CR and FGAs?…”

Section: Introductionmentioning

confidence: 99%

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Power Iterated Color Refinement

Kersting

Mladenov

Garnett

et al. 2014

AAAI

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Color refinement is a basic algorithmic routine for graph isomorphismtesting and has recently been used for computing graph kernels as well as for lifting belief propagation and linear programming. So far, color refinement has been treated as a combinatorial problem. Instead, we treat it as a nonlinear continuous optimization problem and prove thatit implements a conditional gradient optimizer that can be turned into graph clustering approaches using hashing and truncated power iterations. This shows that color refinement is easy to understand in terms of random walks, easy to implement (matrix-matrix/vector multiplications) and readily parallelizable. We support our theoretical results with experiments on real-world graphs with millions of edges.

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Section: Discussionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%