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A fast new algorithm for weak graph regularity

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  03 May 2019

Jacob Fox ,
László Miklós Lovász  and
Yufei Zhao
Jacob Fox
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA
László Miklós Lovász*
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA
Yufei Zhao
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA
*
*Corresponding author. Email: lmlovasz@mit.edu
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Abstract

We provide a deterministic algorithm that finds, in ɛ-O(1)n2 time, an ɛ-regular Frieze–Kannan partition of a graph on n vertices. The algorithm outputs an approximation of a given graph as a weighted sum of ɛ-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 ɛnv(H), in time ɛ-OH(1)n2.

MSC classification

Primary: 05C85: Graph algorithms
Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 05D99: None of the above, but in this section
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 5 , September 2019 , pp. 777 - 790
Copyright
© Cambridge University Press 2019 

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Footnotes

†

J. Fox is supported by a Packard Fellowship, by NSF CAREER award DMS 1352121, and by an Alfred P. Sloan Fellowship.

‡

L. M. Lovász is supported by NSF Postdoctoral Fellowship Award DMS 1705204.

§

Y. Zhao is supported by NSF awards DMS-1362326 and DMS-1764176, and the MIT Solomon Buchsbaum Fund.

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