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On Regularity Lemmas and their Algorithmic Applications

  • JACOB FOX (a1), LÁSZLÓ MIKLÓS LOVÁSZ (a2) and YUFEI ZHAO (a3)
  • Please note a correction has been issued for this article.
Abstract

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition with k parts if there exists an ε-regular partition with k parts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

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[1] Alon, N., Duke, R. A., Lefmann, H., Rödl, V. and Yuster, R. (1994) The algorithmic aspects of the regularity lemma. J. Algorithms 16 80109.
[2] Alon, N. and Naor, A. (2006) Approximating the cut-norm via Grothendieck's inequality. SIAM J. Comput. 35 787803.
[3] Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, third edition, Wiley.
[4] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008) Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. Adv. Math. 219 18011851.
[5] Conlon, D. and Fox, J. (2012) Bounds for graph regularity and removal lemmas. Geom. Funct. Anal. 22 11911256.
[6] Cooper, J. N. (2006) A permutation regularity lemma. Electron. J. Combin. 13 22.
[7] Coppersmith, D. and Winograd, S. (1990) Matrix multiplication via arithmetic progressions. J. Symbol. Comput. 9 251280.
[8] Dellamonica, D., Kalyanasundaram, S., Martin, D., Rödl, V. and Shapira, A. (2012) A deterministic algorithm for the Frieze–Kannan regularity lemma. SIAM J. Discrete Math. 26 1529.
[9] Dellamonica, D. Jr, Kalyanasundaram, S., Martin, D. M., Rödl, V. and Shapira, A. (2015) An optimal algorithm for finding Frieze–Kannan regular partitions. Combin. Probab. Comput. 24 407437.
[10] Duke, R. A., Lefmann, H. and Rödl, V. (1995) A fast approximation algorithm for computing the frequencies of subgraphs in a given graph. SIAM J. Comput. 24 598620.
[11] Fischer, E., Matsliah, A. and Shapira, A. (2010) Approximate hypergraph partitioning and applications. SIAM J. Comput. 39 31553185.
[12] Fox, J. and Lovász, L. M. A tight lower bound for Szemerédi's regularity lemma. Combinatorica, to appear.
[13] Frieze, A. and Kannan, R. (1999) Quick approximation to matrices and applications. Combinatorica 19 175220.
[14] Frieze, A. and Kannan, R. (1999) A simple algorithm for constructing Szemerédi's regularity partition. Electron. J. Combin. 6 17.
[15] Gowers, W. T. (1997) Lower bounds of tower type for Szemerédi's uniformity lemma. Geom. Funct. Anal. 7 322337.
[16] Håstad, J. (1999) Clique is hard to approximate within n 1-ε . Acta Mathematica 182 105142.
[17] Hoppen, C., Kohayakawa, Y. and Sampaio, R. M. (2012) A note on permutation regularity. Discrete Appl. Math. 160 27162727.
[18] Kohayakawa, Y., Rödl, V. and Thoma, L. (2003) An optimal algorithm for checking regularity. SIAM J. Comput. 32 12101235.
[19] Komlós, J. and Simonovits, M. (1996) Szemerédi's regularity lemma and its applications in graph theory. In Combinatorics: Paul Erdős is Eighty, Vol. 2, János Bolyai Mathematical Society, pp. 295352.
[20] Le Gall, F. (2014) Powers of tensors and fast matrix multiplication. In ISSAC '14: Proc. 39th International Symposium on Symbolic and Algebraic Computation, ACM, pp. 296303.
[21] Lovász, L. (2012) Large Networks and Graph Limits, Vol. 60 of American Mathematical Society Colloquium Publications, AMS.
[22] Lovász, L. and Szegedy, B. (2007) Szemerédi's lemma for the analyst. Geom. Funct. Anal. 17 252270.
[23] Moshkovitz, G. and Shapira, A. (2016) A short proof of Gowers' lower bound for the regularity lemma. Combinatorica 36 187194.
[24] Szemerédi, E. (1975) On sets of integers containing no k elements in arithmetic progression. In Proc. International Congress of Mathematicians 1974, Vol. 2, Canadian Mathematical Congress, pp. 503505.
[25] Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes, Vol. 260 of Colloq. Internat. CNRS, CNRS, pp. 399401.
[26] Tao, T. (2010) An Epsilon of Room, II, AMS.
[27] Zuckerman, D. (2007) Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3 103128.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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