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Graph Partitioning via Adaptive Spectral Techniques

Published online by Cambridge University Press:  13 November 2009

AMIN COJA-OGHLAN
AMIN COJA-OGHLAN*
Affiliation:
School of Informatics, University of Edinburgh, Crichton Street, Edinburgh EH8 9AB, UK (e-mail: acoghlan@inf.ed.ac.uk)
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Abstract

In this paper we study the use of spectral techniques for graph partitioning. Let G = (V, E) be a graph whose vertex set has a ‘latent’ partition V1,. . ., Vk. Moreover, consider a ‘density matrix’ Ɛ = (Ɛvw)v, sw∈V such that, for v ∈ Vi and w ∈ Vj, the entry Ɛvw is the fraction of all possible Vi−Vj-edges that are actually present in G. We show that on input (G, k) the partition V1,. . ., Vk can (very nearly) be recovered in polynomial time via spectral methods, provided that the following holds: Ɛ approximates the adjacency matrix of G in the operator norm, for vertices v ∈ Vi, w ∈ Vj ≠ Vi the corresponding column vectors Ɛv, Ɛw are separated, and G is sufficiently ‘regular’ with respect to the matrix Ɛ. This result in particular applies to sparse graphs with bounded average degree as n = #V → ∞, and it has various consequences on partitioning random graphs.

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 2 , March 2010 , pp. 227 - 284
Copyright
Copyright © Cambridge University Press 2009

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