The algorithmic aspects of the regularity lemma
Journal of Algorithms
A Fast Approximation Algorithm for Computing theFrequencies of Subgraphs in a Given Graph
SIAM Journal on Computing
MAX-CUT has a randomized approximation scheme in dense graphs
Random Structures & Algorithms
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Polynomial time approximation schemes for dense instances of NP -hard problems
Journal of Computer and System Sciences
Hypergraphs, quasi-randomness, and conditions for regularity
Journal of Combinatorial Theory Series A
An Algorithmic Regularity Lemma for Hypergraphs
SIAM Journal on Computing
Property testers for dense constraint satisfaction programs on finite domains
Random Structures & Algorithms
The regularity lemma and approximation schemes for dense problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
An Optimal Algorithm for Checking Regularity
SIAM Journal on Computing
Random sampling and approximation of MAX-CSPs
Journal of Computer and System Sciences - STOC 2002
Regularity lemma for k-uniform hypergraphs
Random Structures & Algorithms
The counting lemma for regular k-uniform hypergraphs
Random Structures & Algorithms
Approximating the Cut-Norm via Grothendieck's Inequality
SIAM Journal on Computing
A variant of the hypergraph removal lemma
Journal of Combinatorial Theory Series A
Testing versus Estimation of Graph Properties
SIAM Journal on Computing
An Algorithmic Version of the Hypergraph Regularity Method
SIAM Journal on Computing
Szemerédi-type clustering of peer-to-peer streaming system
Proceedings of the 2011 International Workshop on Modeling, Analysis, and Control of Complex Networks
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Szemerédi's regularity lemma is a cornerstone result in extremal combinatorics. It (roughly) asserts that any dense graph is composed of a finite number of pseudorandom graphs. The regularity lemma has found many applications in theoretical computer science, and thus a lot of attention was given to designing algorithmic versions of this lemma. Our main results in this paper are the following: (i) We introduce a new approach to the problem of constructing regular partitions of graphs, which results in a surprisingly simple $O(n)$ time algorithmic version of the regularity lemma, thus improving over the previous $O(n^2)$ time algorithms. Furthermore, unlike all the previous approaches for this problem (see [N. Alon and A. Naor, SIAM J. Comput., 35 (2006), pp. 787-803], [R. A. Duke, H. Lefmann, and V. Rödl, SIAM J. Comput., 24 (1995), pp. 598-620], [A. Frieze and R. Kannan, Electron. J. Combin., 6 (1999), article 17], [A. Frieze and R. Kannan, “The regularity lemma and approximation schemes for dense problems,” in Proceedings of the 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 12-20], and [Y. Kohayakawa, V. Rödl, and L. Thoma, SIAM J. Comput., 32 (2003), pp. 1210-1235]), which only guaranteed to find tower-size partitions, our algorithm will find a small regular partition, if one exists in the graph. (ii) For any constant $r\geq3$ we give an $O(n)$ time randomized algorithm for constructing regular partitions of $r$-uniform hypergraphs, thus improving the previous $O(n^{2r-1})$ time (deterministic) algorithms [A. Czygrinow and V. Rödl, SIAM J. Comput., 30 (2000), pp. 1041-1066], [A. Frieze and R. Kannan, “The regularity lemma and approximation schemes for dense problems,” in Proceedings of the 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 12-20]. These two results are obtained as an application of an efficient algorithm for approximating partition problems of hypergraphs which we obtain here: Given a (directed) hypergraph with bounded edge arities, a set of constraints on the set sizes and densities of a possible partition of its vertex set, and an approximation parameter, we provide in $O(n)$ time a partition approximating the constraints if a partition satisfying them exists. We can also test in $O(1)$ time for the existence of such a partition given the approximation parameter. This algorithm extends the result of Goldreich, Goldwasser, and Ron for graph partition problems [O. Goldreich, S. Goldwasser, and D. Ron, J. ACM, 45 (1998), pp. 653-750] and encompasses more recent hypergraph-related results such as the maximal constraint satisfaction approximation of [G. Andersson and L. Engebretsen, Random Structures Algorithms, 21 (2002), pp. 14-32].