Skip to main content Accessibility help
×
Home
Hostname: page-component-747cfc64b6-bv7lh Total loading time: 0.356 Render date: 2021-06-17T06:55:44.761Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

On Regularity Lemmas and their Algorithmic Applications

Published online by Cambridge University Press:  28 March 2017

JACOB FOX
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail: jacobfox@stanford.edu)
LÁSZLÓ MIKLÓS LOVÁSZ
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (e-mail: lmlovasz@math.mit.edu)
YUFEI ZHAO
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK (e-mail: yufei.zhao@maths.ox.ac.uk)

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.

Type
Paper
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below.

References

[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.CrossRefGoogle Scholar
[2] Alon, N. and Naor, A. (2006) Approximating the cut-norm via Grothendieck's inequality. SIAM J. Comput. 35 787803.CrossRefGoogle Scholar
[3] Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, third edition, Wiley.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[5] Conlon, D. and Fox, J. (2012) Bounds for graph regularity and removal lemmas. Geom. Funct. Anal. 22 11911256.CrossRefGoogle Scholar
[6] Cooper, J. N. (2006) A permutation regularity lemma. Electron. J. Combin. 13 22.Google Scholar
[7] Coppersmith, D. and Winograd, S. (1990) Matrix multiplication via arithmetic progressions. J. Symbol. Comput. 9 251280.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[11] Fischer, E., Matsliah, A. and Shapira, A. (2010) Approximate hypergraph partitioning and applications. SIAM J. Comput. 39 31553185.CrossRefGoogle Scholar
[12] Fox, J. and Lovász, L. M. A tight lower bound for Szemerédi's regularity lemma. Combinatorica, to appear.Google Scholar
[13] Frieze, A. and Kannan, R. (1999) Quick approximation to matrices and applications. Combinatorica 19 175220.CrossRefGoogle Scholar
[14] Frieze, A. and Kannan, R. (1999) A simple algorithm for constructing Szemerédi's regularity partition. Electron. J. Combin. 6 17.Google Scholar
[15] Gowers, W. T. (1997) Lower bounds of tower type for Szemerédi's uniformity lemma. Geom. Funct. Anal. 7 322337.CrossRefGoogle Scholar
[16] Håstad, J. (1999) Clique is hard to approximate within n 1-ε . Acta Mathematica 182 105142.CrossRefGoogle Scholar
[17] Hoppen, C., Kohayakawa, Y. and Sampaio, R. M. (2012) A note on permutation regularity. Discrete Appl. Math. 160 27162727.CrossRefGoogle Scholar
[18] Kohayakawa, Y., Rödl, V. and Thoma, L. (2003) An optimal algorithm for checking regularity. SIAM J. Comput. 32 12101235.CrossRefGoogle Scholar
[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.Google Scholar
[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.CrossRefGoogle Scholar
[21] Lovász, L. (2012) Large Networks and Graph Limits, Vol. 60 of American Mathematical Society Colloquium Publications, AMS.Google Scholar
[22] Lovász, L. and Szegedy, B. (2007) Szemerédi's lemma for the analyst. Geom. Funct. Anal. 17 252270.CrossRefGoogle Scholar
[23] Moshkovitz, G. and Shapira, A. (2016) A short proof of Gowers' lower bound for the regularity lemma. Combinatorica 36 187194.CrossRefGoogle Scholar
[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.Google Scholar
[25] Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes, Vol. 260 of Colloq. Internat. CNRS, CNRS, pp. 399401.Google Scholar
[26] Tao, T. (2010) An Epsilon of Room, II, AMS.Google Scholar
[27] Zuckerman, D. (2007) Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3 103128.CrossRefGoogle Scholar
6
Cited by

Linked content

Please note a has been issued for this article.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On Regularity Lemmas and their Algorithmic Applications
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On Regularity Lemmas and their Algorithmic Applications
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On Regularity Lemmas and their Algorithmic Applications
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *