A deterministic algorithm for the Frieze-Kannan regularity lemma

Domingos Dellamonica; Subrahmanyam Kalyanasundaram; Daniel Martin; Vojtěch Rödl; Asaf Shapira

doi:10.1007/978-3-642-22935-0_42

A deterministic algorithm for the Frieze-Kannan regularity lemma

Domingos Dellamonica, Subrahmanyam Kalyanasundaram, Daniel Martin, Vojtěch Rödl, Asaf Shapira

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

The Frieze-Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. Williams [24] recently asked if one can construct a partition satisfying the conditions of the Frieze-Kannan regularity lemma in deterministic sub-cubic time. We resolve this problem by designing an Õ(n^ω) time algorithm for constructing such a partition, where ω < 2.376 is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze-Kannan regularity lemma.

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization
Subtitle of host publication	Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings
Pages	495-506
Number of pages	12
DOIs	https://doi.org/10.1007/978-3-642-22935-0_42
State	Published - 2011
Externally published	Yes
Event	14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011 - Princeton, NJ, United States Duration: 17 Aug 2011 → 19 Aug 2011

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	6845 LNCS

Conference

Conference	14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011
Country/Territory	United States
City	Princeton, NJ
Period	17/08/11 → 19/08/11

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-642-22935-0_42

Cite this

Dellamonica, D., Kalyanasundaram, S., Martin, D., Rödl, V., & Shapira, A. (2011). A deterministic algorithm for the Frieze-Kannan regularity lemma. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings (pp. 495-506). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6845 LNCS). https://doi.org/10.1007/978-3-642-22935-0_42

Dellamonica, Domingos ; Kalyanasundaram, Subrahmanyam ; Martin, Daniel et al. / A deterministic algorithm for the Frieze-Kannan regularity lemma. Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings. 2011. pp. 495-506 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "A deterministic algorithm for the Frieze-Kannan regularity lemma",

abstract = "The Frieze-Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. Williams [24] recently asked if one can construct a partition satisfying the conditions of the Frieze-Kannan regularity lemma in deterministic sub-cubic time. We resolve this problem by designing an {\~O}(nω) time algorithm for constructing such a partition, where ω < 2.376 is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze-Kannan regularity lemma.",

author = "Domingos Dellamonica and Subrahmanyam Kalyanasundaram and Daniel Martin and Vojt{\v e}ch R{\"o}dl and Asaf Shapira",

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booktitle = "Approximation, Randomization, and Combinatorial Optimization",

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Dellamonica, D, Kalyanasundaram, S, Martin, D, Rödl, V & Shapira, A 2011, A deterministic algorithm for the Frieze-Kannan regularity lemma. in Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6845 LNCS, pp. 495-506, 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011, Princeton, NJ, United States, 17/08/11. https://doi.org/10.1007/978-3-642-22935-0_42

A deterministic algorithm for the Frieze-Kannan regularity lemma. / Dellamonica, Domingos; Kalyanasundaram, Subrahmanyam; Martin, Daniel et al.
Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings. 2011. p. 495-506 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6845 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - The Frieze-Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. Williams [24] recently asked if one can construct a partition satisfying the conditions of the Frieze-Kannan regularity lemma in deterministic sub-cubic time. We resolve this problem by designing an Õ(nω) time algorithm for constructing such a partition, where ω < 2.376 is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze-Kannan regularity lemma.

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Dellamonica D, Kalyanasundaram S, Martin D, Rödl V, Shapira A. A deterministic algorithm for the Frieze-Kannan regularity lemma. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings. 2011. p. 495-506. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-22935-0_42

A deterministic algorithm for the Frieze-Kannan regularity lemma

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All Science Journal Classification (ASJC) codes

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