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Quasi-randomness and algorithmic regularity for graphs with general degree distributions

By Noga Alon, Amin Coja-Oghlan, Hiêp Hàn, Mihyun Kang, Vojtěch Rödl and Mathias Schacht

Abstract

We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to express how much a given graph “resembles” a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we introduce a concept of regularity that takes into account vertex weights, and show that if $G=(V,E)$ satisfies a certain boundedness condition, then $G$ admits a regular partition. In addition, building on the work of Alon and Naor [Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago, IL, ACM, New York, 2004, pp. 72–80], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph $G$ in polynomial time. As an application, we present a polynomial time approximation scheme for MAX CUT on (sparse) graphs without “dense spots.

Topics: QA
Publisher: Society for Industrial and Applied Mathematics
Year: 2010
OAI identifier: oai:wrap.warwick.ac.uk:3320

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