Testing versus estimation of graph properties

Eldar Fischer, Ilan Newman

Research output: Contribution to journalArticlepeer-review


Tolerant testing is an emerging topic in the field of property testing, which was defined in [M. Parnas, D. Ron, and R. Rubinfeld, J. Comput. System Sci., 72 (2006), pp. 1012-1042] and has recently become a very active topic of research. In the general setting, there exist properties that are testable but are not tolerantly testable [E. Fischer and L. Fortnow, Proceedings of the 20th IEEE Conference on Computational Complexity, 2005, pp. 135-140]. On the other hand, we show here that in the setting of the dense graph model, all testable properties are not only tolerantly testable (which was already implicitly proved in [N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy, Combinatorica, 20 (2000), pp. 451-476] and [O. Goldreich and L. Trevisan, Random Structures Algorithms, 23 (2003), pp. 23-57]), but also admit a constant query size algorithm that estimates the distance from the property up to any fixed additive constant. In the course of the proof we develop a framework for extending Szemerédi's regularity lemma, both as a prerequisite for formulating what kind of information about the input graph will provide us with the correct estimation, and as the means for efficiently gathering this information. In particular, we construct a probabilistic algorithm that finds the parameters of a regular partition of an input graph using a constant number of queries, and an algorithm to find a regular partition of a graph using a TC0 circuit. This, in some ways, strengthens the results of [N. Alon, R. A. Duke, H. Lefmann, V. Rödl, and R. Yuster, J. Algorithms, 16 (1994), pp. 80-109].

Original languageEnglish
Pages (from-to)482-501
Number of pages20
JournalSIAM Journal on Computing
Issue number2
StatePublished - 2007


  • Approximation
  • Graph algorithms
  • Property testing
  • Regularity lemma

ASJC Scopus subject areas

  • General Computer Science
  • General Mathematics


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