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An Optimal Algorithm for Finding Frieze–Kannan Regular Partitions

Published online by Cambridge University Press:  23 April 2014

DOMINGOS DELLAMONICA Jr ,
SUBRAHMANYAM KALYANASUNDARAM ,
DANIEL M. MARTIN ,
VOJTĚCH RÖDL  and
ASAF SHAPIRA
DOMINGOS DELLAMONICA Jr
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA (e-mail: ddellam@mathcs.emory.edu)
SUBRAHMANYAM KALYANASUNDARAM
Affiliation:
Department of Computer Science and Engineering, IIT Hyderabad, India (e-mail: subruk@iith.ac.in)
DANIEL M. MARTIN
Affiliation:
Center for Mathematics, Computer Science and Cognition, Universidade Federal do ABC, Santo André, SP 09210-170Brazil (e-mail: daniel.martin@ufabc.edu.br)
VOJTĚCH RÖDL
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA (e-mail: rodl@mathcs.emory.edu)
ASAF SHAPIRA
Affiliation:
School of Mathematics, Tel Aviv University, Tel Aviv, Israel69978 (e-mail: asafico@tau.ac.il)
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Abstract

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In this paper we prove that two local conditions involving the degrees and co-degrees in a graph can be used to determine whether a given vertex partition is Frieze–Kannan regular. With a more refined version of these two local conditions we provide a deterministic algorithm that obtains a Frieze–Kannan regular partition of any graph G in time O(|V(G)|2).

Keywords

Primary 05C35Secondary 05C4205C50

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 2 , March 2015 , pp. 407 - 437
Copyright
Copyright © Cambridge University Press 2014 

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 80–109.Google Scholar
[2]Conlon, D. and Fox, J. (2012) Bounds for graph regularity and removal lemmas. Geom. Funct. Anal. 22 1191–1256.Google Scholar
[3]Coppersmith, D. and Winograd, S. (1990) Matrix multiplication via arithmetic progressions. J. Symbol. Comput. 9 251–280.Google Scholar
[4]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 15–29.Google Scholar
[5]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. 24 598–620.Google Scholar
[6]Erdős, P. and Turán, P. (1936) On some sequences of integers. J. London Math. Soc. (S1) 11 261–264.Google Scholar
[7]Frieze, A. and Kannan, R. (1999) Quick approximation to matrices and applications. Combinatorica 19 175–220.Google Scholar
[8]Frieze, A. and Kannan, R. (1996) The regularity lemma and approximation schemes for dense problems. In 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Computer Society Press, pp. 12–20.Google Scholar
[9]Gowers, W. T. (1997) Lower bounds of tower type for Szemerédi's uniformity lemma. Geom. Funct. Anal. 7 322–337.Google Scholar
[10]Kohayakawa, Y., Rödl, V. and Thoma, L. (2003) An optimal algorithm for checking regularity. SIAM J. Comput. 32 1210–1235.CrossRefGoogle Scholar
[11]Schacht, M. and Rödl, V. (2010) Fete of combinatorics and computer science. In Regularity Lemmas for Graphs (Katona, G., Schrijver, A. and Szönyi, T., eds), Springer, pp. 287–326.Google Scholar
[12]Szemerédi, E. (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27 199–245.Google Scholar
[13]Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes (Orsay 1976), Vol. 260 of Colloques Internationaux du CNRS, CNRS, pp. 399–401.Google Scholar
[14]Williams, R. (2009) Private communication.Google Scholar