Algorithmic Aspects of Property Testing in the Dense Graphs Model

  • Oded Goldreich
  • Dana Ron
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5687)

Abstract

In this paper we consider two basic questions regarding the query complexity of testing graph properties in the adjacency matrix model. The first question refers to the relation between adaptive and non-adaptive testers, whereas the second question refers to testability within complexity that is inversely proportional to the proximity parameter, denoted ε. The study of these questions reveals the importance of algorithmic design (also) in this model. The highlights of our study are:

  • A gap between the complexity of adaptive and non-adaptive testers. Specifically, there exists a (natural) graph property that can be tested using \({\widetilde{O}}(\epsilon^{-1})\) adaptive queries, but cannot be tested using o(ε − 3/2) non-adaptive queries.

  • In contrast, there exist natural graph properties that can be tested using \({\widetilde{O}}(\epsilon^{-1})\) non-adaptive queries, whereas Ω(ε − 1) queries are required even in the adaptive case.

We mention that the properties used in the foregoing conflicting results have a similar flavor, although they are of course different.

Keywords

Bipartite Graph Query Complexity Complete Bipartite Graph Graph Property Adaptive Tester 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N.: On the number of subgraphs of prescribed type of graphs with a given number of edges. Israel J. Math. 38, 116–130 (1981)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alon, N., Fischer, E., Krivelevich, M., Szegedy, M.: Efficient Testing of Large Graphs. Combinatorica 20, 451–476 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alon, N., Fischer, E., Newman, I.: Testing of bipartite graph properties. SIAM Journal on Computing 37, 959–976 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alon, N., Fischer, E., Newman, I., Shapira, A.: A Combinatorial Characterization of the Testable Graph Properties: It’s All About Regularity. In: 38th STOC, pp. 251–260 (2006)Google Scholar
  5. 5.
    Alon, N., Shapira, A.: A Characterization of Easily Testable Induced Subgraphs. Combinatorics Probability and Computing 15, 791–805 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ben-Sasson, E., Harsha, P., Raskhodnikova, S.: 3CNF properties are hard to test. SIAM Journal on Computing 35(1), 1–21 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bogdanov, A., Trevisan, L.: Lower Bounds for Testing Bipartiteness in Dense Graphs. In: IEEE Conference on Computational Complexity, pp. 75–81 (2004)Google Scholar
  8. 8.
    Canetti, R., Even, G., Goldreich, O.: Lower Bounds for Sampling Algorithms for Estimating the Average. IPL 53, 17–25 (1995)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fischer, F.E.: The art of uninformed decisions: A primer to property testing. Bulletin of the European Association for Theoretical Computer Science 75, 97–126 (2001)MATHGoogle Scholar
  10. 10.
    Fischer, E.: On the strength of comparisons in property testing. Inform. and Comput. 189(1), 107–116 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. Journal of the ACM, 653–750 (July 1998)Google Scholar
  12. 12.
    Goldreich, O., Ron, D.: Property Testing in Bounded Degree Graphs. Algorithmica 32(2), 302–343 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Goldreich, O., Ron, D.: Algorithmic Aspects of Property Testing in the Dense Graphs Model. ECCC, TR08-039 (2008)Google Scholar
  14. 14.
    Goldreich, O., Ron, D.: On Proximity Oblivious Testing. ECCC, TR08-041 (2008); Extended abstract in the proceedings of the 41st STOC (2009)Google Scholar
  15. 15.
    Goldreich, O., Trevisan, L.: Three theorems regarding testing graph properties. Random Structures and Algorithms 23(1), 23–57 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gonen, M., Ron, D.: On the Benefit of Adaptivity in Property Testing of Dense Graphs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 525–539. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Ron, D.: Property testing. In: Rajasekaran, S., Pardalos, P.M., Reif, J.H., Rolim, J.D.P. (eds.) Handbook on Randomization, vol. II, pp. 597–649 (2001)Google Scholar
  18. 18.
    Raskhodnikova, S., Smith, A.: A note on adaptivity in testing properties of bounded-degree graphs. ECCC, TR06-089 (2006)Google Scholar
  19. 19.
    Rubinfeld, R.R., Sudan, M.: Robust characterization of polynomials with applications to program testing. SIAM Journal on Computing 25(2), 252–271 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Oded Goldreich
    • 1
  • Dana Ron
    • 2
  1. 1.Department of Computer ScienceWeizmann Institute of ScienceRehovotIsrael
  2. 2.Department of Electrical Engineering-SystemsTel Aviv UniversityTel AvivIsrael

Personalised recommendations