The algorithmic aspects of the regularity lemma
Journal of Algorithms
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)
Statistical mechanics of complex networks
Statistical mechanics of complex networks
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
An efficient sparse regularity concept
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
An Efficient Sparse Regularity Concept
SIAM Journal on Discrete Mathematics
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We deal with two very related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to measure 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 present a novel concept of regularity that takes into account the graph's degree distribution, 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 [4], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph G in polynomial time.