A fast new algorithm for weak graph regularity

01/12/2018

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by   Jacob Fox, et al.

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We provide a deterministic algorithm that finds, in ϵ^-O(1) n^2 time, an ϵ-regular Frieze-Kannan partition of a graph on n vertices. The algorithm outputs an approximation of a given graph as a weighted sum of ϵ^-O(1) many complete bipartite graphs. As a corollary, we give a deterministic algorithm for estimating the number of copies of H in an n-vertex graph G up to an additive error of at most ϵ n^v(H), in time ϵ^-O_H(1)n^2.