Abstract
Focusing on property testing tasks that have query complexity that is independent of the size of the tested object (i.e., depends on the proximity parameter only), we prove the existence of a rich hierarchy of the corresponding complexity classes. That is, for essentially any function \(q : (0, 1] \rightarrow \mathbb{N}\), we prove the existence of properties for which \(\epsilon\)-testing has query complexity \(\Theta(q(\Theta(\epsilon)))\). Such results are proved in three standard domains that are often considered in property testing: generic functions, adjacency predicates describing (dense) graphs, and incidence functions describing bounded-degree graphs.
These results complement hierarchy theorems of Goldreich, Krivelevich, Newman, and Rozenberg (Computational Complexity, 2012), which refer to the dependence of the query complexity on the size of the tested object, and focus on the case that the proximity parameter is set to some small positive constant. We actually combine both flavors and get tight results on the query complexity of testing when allowing the query complexity to depend on both the size of the object and the proximity parameter.
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References
Alon, N.: Testing subgraphs of large graphs. Random Structures and Algorithms, Vol. 21, 359–370 (2002)
N. Alon, E. Fischer, I. Newman, and A. Shapira. A Combinatorial Characterization of the Testable Graph Properties: It's All About Regularity. In 38th STOC, pages 251–260, 2006
Alon, N., Shapira, A.: Testing subgraphs in directed graphs. JCSS, Vol. 69, 354–482 (2004)
Alon, N., Shapira, A.: Every Monotone Graph Property is Testable. SIAM Journal on Computing, Vol. 38, 505–522 (2008)
I. Benjamini, O. Schramm, and A. Shapira. Every Minor-Closed Property of Sparse Graphs is Testable. In 40th STOC, pages 393–402, 2008
Blum, M., Luby, M., Rubinfeld, R.: Self-Testing/Correcting with Applications to Numerical Problems. JCSS, Vol. 47(3), 549–595 (1993)
A. Bogdanov, K. Obata, and L. Trevisan. A lower bound for testing 3-colorability in bounded-degree graphs. In 43rd FOCS, pages 93–102, 2002
L. Gishboliner and A. Shapira. A Generalized Turan Problem and its Applications. ECCC, TR18-007, 2018
O. Goldreich. Introduction to Property Testing. Cambridge University Press, 2017.
O. Goldreich. Flexible models for testing graph properties. ECCC, TR18-104, 2018
O. Goldreich. Testing Graphs in Vertex-Distribution-Free Models. ECCC, TR18-171, 2018
O. Goldreich, S. Goldwasser, and D. Ron. Property testing and its connection to learning and approximation. Journal of the ACM, pages 653–750, July 1998
Goldreich, O., Krivelevich, M., Newman, I., Rozenberg, E.: Hierarchy Theorems for Property Testing. Computational Complexity, Vol. 21(1), 129–192 (2012)
Goldreich, O., Ron, D.: Property Testing in Bounded Degree Graphs. Algorithmica, Vol. 32(2), 302–343 (2002)
Rubinfeld, R., Sudan, M.: Robust characterization of polynomials with applications to program testing. SIAM Journal on Computing 25(2), 252–271 (1996)
Acknowledgements
This research was partially supported by the Israel Science Foundation (grants No. 671/13 and 1146/18).
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Goldreich, O. Hierarchy Theorems for Testing Properties in Size-Oblivious Query Complexity. comput. complex. 28, 709–747 (2019). https://doi.org/10.1007/s00037-019-00187-2
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DOI: https://doi.org/10.1007/s00037-019-00187-2