Skip to main content
SpringerLink
Log in

Efficient Testing of Hypergraphs

Extended Abstract

  • Conference paper
  • First Online:
Automata, Languages and Programming (ICALP 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2380))

Included in the following conference series:

Abstract

We investigate a basic problem in combinatorial property testing, in the sense of Goldreich, Goldwasser, and Ron [9,10], in the context of 3-uniform hypergraphs, or 3-graphs for short. As customary, a 3-graph F is simply a collection of 3-element sets. Let Forbind(n, F) be the family of all 3-graphs on n vertices that contain no copy of F as an induced subhypergraph. We show that the property “H ∈ Forbind(n, F)” is testable, for any 3-graph F. In fact, this is a consequence of a new, basic combinatorial lemma, which extends to 3-graphs a result for graphs due to Alon, Fischer, Krivelevich, and Szegedy [2,3].

Indeed, we prove that if more than ξn 3 (V > 0) triples must be added or deleted from a 3-graph H on n vertices to destroy all induced copies of F, then H must contain ≥ cn |V(F)| induced copies of F, as long as n ≥ n 0(ξ, F). Our approach is inspired in [2,3], but the main ingredients are recent hypergraph regularity lemmas and counting lemmas for 3-graphs.

Partially supported by MCT/CNPq through ProNEx Programme (Proc. CNPq 664107/1997-4), by CNPq (Proc. 300334/93-1, 910064/99-7, and 468516/2000-0).

Partially supported by NSF Grant INT-0072064.

Partially supported by NSF Grants 0071261 and INT-0072064.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. Alon, R. A. Duke, H. Lefmann, V. Rödl, and R. Yuster, The algorithmic aspects of the regularity lemma, J. Algorithms 16 (1994), no. 1, 80–109.

    Article  MATH  MathSciNet  Google Scholar 

  2. N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy, Efficient testing of large graphs (extended abstract), 40th Annual Symposium on Foundations of Computer Science (New York City, NY), IEEE Comput. Soc. Press, 1999, pp. 656–666.

    Google Scholar 

  3. -, Efficient testing of large graphs, Combinatorica 20 (2000), no. 4, 451–476.

    Article  MATH  MathSciNet  Google Scholar 

  4. Noga Alon, Testing subgraphs in large graphs, Proceedings of the 42nd IEEE Annual Symposium on Foundations of Computer Science (FOCS 2001), 2001, pp. 434–439.

    Google Scholar 

  5. J. Bourgain, On triples in arithmetic progression, Geom. Funct. Anal. 9 (1999), no. 5, 968–984.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Czumaj and C. Sohler, Testing hypergraph coloring, Proc. of ICALP, 2001, pp. 493–505.

    Google Scholar 

  7. Peter Frankl and Vojtěch Rödl, Extremal problems on set systems, Random Structures and Algorithms 20 (2002), no. 2, 131–164.

    Article  MATH  MathSciNet  Google Scholar 

  8. Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256.

    Article  MATH  MathSciNet  Google Scholar 

  9. Oded Goldreich, Shafi Goldwasser, and Dana Ron, Property testing and its connection to learning and approximation, 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Comput. Soc. Press, Los Alamitos, CA, 1996, pp. 339–348.

    Google Scholar 

  10. -, Property testing and its connection to learning and approximation, Journal of the Association for Computing Machinery 45 (1998), no. 4, 653–750.

    MATH  MathSciNet  Google Scholar 

  11. W. T. Gowers, A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), no. 3, 529–551.

    Article  MATH  MathSciNet  Google Scholar 

  12. -, A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), no. 3, 465–588.

    Google Scholar 

  13. Y. Kohayakawa, B. Nagle, and V. Rödl, Testing hypergraphs, In preparation, 2002.

    Google Scholar 

  14. Brendan Nagle and Vojtěch Rödl, Regularity properties for triple systems, Random Structures & Algorithms (2002), to appear.

    Google Scholar 

  15. Vojtečh Rödl and Jozef Skokan, Uniformity of set systems, manuscript, 2001.

    Google Scholar 

  16. Dana Ron, Property testing, Handbook of randomized algorithms (P. M. Pardalos, S. Rajasekaran, J. Reif, and J. D. P. Rolim, eds.), Kluwer Academic Publishers, 2001, to appear.

    Google Scholar 

  17. K. F. Roth, On certain sets of integers, Journal of the London Mathematical Society 28 (1953), 104–109.

    Article  MATH  Google Scholar 

  18. -, Irregularities of sequences relative to arithmetic progressions. IV, Period. Math. Hungar. 2 (1972), 301–326, Collection of articles dedicated to the memory of Alfréd Rényi, I.

    Article  MATH  MathSciNet  Google Scholar 

  19. I. Z. Ruzsa and E. Szemerédi, Triple systems with no six points carrying three triangles, Combinatorics (Proceedings of the Fifth Hungarian Colloquium, Keszthely, 1976), Vol. II (Amsterdam-New York), Colloq. Math. Soc. János Bolyai, no. 18, North-Holland, 1978, pp. 939–945.

    Google Scholar 

  20. E. Szemerédi, On sets of integers containing no four elements in arithmetic progression, Acta Mathematica Academiae Scientiarum Hungaricae 20 (1969), 89–104.

    Article  MATH  MathSciNet  Google Scholar 

  21. -, On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975), 199–245, collection of articles in memory of Jurii Vladimirovič Linnik.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo, Brazil

    Yoshiharu Kohayakawa

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Brendan Nagle

  3. Department of Mathematics, University of Nevada, Reno, Nevada, 89557, USA

    Brendan Nagle

  4. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    Vojtěch Rödl

Authors
  1. Yoshiharu Kohayakawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Brendan Nagle
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vojtěch Rödl
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zentrum, ETH Zürich, 8092, Zürich, Switzerland

    Peter Widmayer  & Stephan Eidenbenz  & 

  2. Department of Languages and Sciences of the Computation E.T.S. de Ingeniería Informática, University of Málaga, Campus de Teatinos, 29071, Málaga, Spain

    Francisco Triguero , Rafael Morales  & Ricardo Conejo ,  & 

  3. School of Cognitive and Computing Sciences, University of Sussex, Falmer, Brighton, BN1 9QN, UK

    Matthew Hennessy

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kohayakawa, Y., Nagle, B., Rödl, V. (2002). Efficient Testing of Hypergraphs. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds) Automata, Languages and Programming. ICALP 2002. Lecture Notes in Computer Science, vol 2380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45465-9_87

Download citation

  • DOI: https://doi.org/10.1007/3-540-45465-9_87

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43864-9

  • Online ISBN: 978-3-540-45465-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation