The algorithmic aspects of the regularity lemma
Journal of Algorithms
Introduction to matrix analysis (2nd ed.)
Introduction to matrix analysis (2nd ed.)
Szemerédi's regularity lemma for sparse graphs
FoCM '97 Selected papers of a conference on Foundations of computational mathematics
Gadgets, Approximation, and Linear Programming
SIAM Journal on Computing
Some optimal inapproximability results
Journal of the ACM (JACM)
An Optimal Algorithm for Checking Regularity
SIAM Journal on Computing
Approximating the cut-norm via Grothendieck's inequality
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Lifts, Discrepancy and Nearly Optimal Spectral Gap
Combinatorica
Quasi-random graphs with given degree sequences
Random Structures & Algorithms
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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.”