Referring to the query complexity of property testing, we prove the existence of a rich hierarchy of corresponding complexity classes. That is, for any relevant function q, we prove the existence of properties that have testing complexity Θ(q). Such results are proven in three standard domains often considered in property testing: generic functions, adjacency predicates describing (dense) graphs, and incidence functions describing bounded-degree graphs. While in two cases, the proofs are quite straightforward, and the techniques employed in the case of the dense graph model seem significantly more involved. Specifically, problems that arise and are treated in the latter case include (1) the preservation of distances between graph under a blow-up operation and (2) the construction of monotone graph properties that have local structure.
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Alon N.: Testing Subgraphs of Large Graphs. Random Structures and Algorithms. 21, 359–370 (2002)
Alon N., Babai L., Itai A.: A fast and Simple Randomized Algorithm for the Maximal Independent Set Problem. J. of Algorithms. 7, 567–583 (1986)
Alon N., Fischer E., Krivelevich M., Szegedy M.: Efficient Testing of Large Graphs. Combinatorica. 20, 451–476 (2000)
N. Alon, E. Fischer, I. Newman & A. Shapira. A Combinatorial Characterization of the Testable Graph Properties: It’s All About Regularity. In 38th STOC, pages 251–260, 2006.
Alon N., Goldreich O., Hastad J., Peralta R.: Simple constructions of almost k-wise independent random variables. Journal of Random structures and Algorithms. 3(3), 354–482 (1992)
Alon N., Shapira A.: Testing Subgraphs in Directed Graphs. JCSS. 69, 354–482 (2004)
Alon N., Shapira A.: A Characterization of Easily Testable Induced Subgraphs. Combinatorics Probability and Computing. 15, 791–805 (2006)
Alon N., Shapira A.: Every Monotone Graph Property is Testable. SIAM Journal on Computing. 38, 505–522 (2008)
L. Avigad & O. Goldreich. Testing Graph Blow-Up. Unpublished manuscript, 2010. Available from http://www.wisdom.weizmann.ac.il/~oded/p_lidor.html.
I. Benjamini, O. Schramm & 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. 47(3), 549–595 (1993)
Ben-Sasson E., Harsha P., Raskhodnikova S.: 3CNF Properties Are Hard to Test. SIAM Journal on Computing. 35(1), 1–21 (2005)
A. Bogdanov, K. Obata & L. Trevisan. A Lower Bound for Testing 3-Colorability in Bounded-Degree Graphs. In 43rd FOCS, pages 93–102, 2002.
C. Borgs, J. Chayes, L. Lovász, V.T. Sós, B. Szegedy & K. Vesztergombi. Graph Limits and Parameter Testing. In 38th STOC, pages 261–270, 2006.
Ergun F., Kannan S., Kumar S.R., Rubinfeld R., Viswanathan M.: Spot-Checkers. JCSS. 60(3), 717–751 (2000)
Fischer E., Matsliah A.: Testing Graph Isomorphism. SIAM J. Comput. 38(1), 207–225 (2008)
E. Fischer & E. Rozenberg. Inflatable Graph Properties and Natural Property Tests. In 15th RANDOM, Springer, LNCS 6845, pages 542–554, 2011.
Goldreich O., Goldwasser S., Lehman E., Ron D., Samorodnitsky A.: Testing Monotonicity. Combinatorica. 20(3), 301–337 (2000)
O.Goldreich, S. Goldwasser & D. Ron. Property testing and its connection to learning and approximation. Journal of the ACM, Vol. 45 (4), pages 653–750, July 1998.
O. Goldreich, M. Krivelevich, I. Newman & E. Rozenberg. Hierarchy Theorems for Property Testing. ECCC, TR08-097, 2008.
Goldreich O., Ron D.: Property Testing in Bounded Degree Graphs. Algorithmica. 32(2), 302–343 (2002)
Goldreich O., Ron D.: A Sublinear Bipartitness Tester for Bounded Degree Graphs. Combinatorica. 19(3), 335–373 (1999)
O. Goldreich & L. Trevisan. Three theorems regarding testing graph properties. Random Structures and Algorithms, Vol. 23 (1), pages 23–57, August 2003.
Lachish O., Newman I., Shapira A.: Space Complexity vs. Query Complexity. Computational Complexity. 17, 70–93 (2008)
Naor J., Naor M.: Small-bias Probability Spaces: Efficient Constructions and Applications. SIAM J. on Computing. 22, 838–856 (1993)
Parnas M., Ron D., Rubinfeld R.: Testing Membership in Parenthesis Laguages. Random Structures and Algorithms. 22(1), 98–138 (2003)
O. Pikhurko. An Analytic Approach to Stability. Manuscript, 2009. Available from http://arxiv.org/abs/0812.0214.
Ron D.: Property Testing: A Learning Theory Perspective. Foundations and Trends in Machine Learning. 1(3), 307–402 (2008)
Ron D.: Algorithmic and Analysis Techniques in Property Testing. Foundations and Trends in TCS. 5(2), 73–205 (2009)
Rubinfeld R., Sudan M.: Robust characterization of polynomials with applications to program testing. SIAM Journal on Computing. 25(2), 252–271 (1996)
R. Shaltiel. Recent Developments in Explicit Constructions of Extractors. In Current Trends in Theoretical Computer Science: The Challenge of the New Century, Vol 1: Algorithms and Complexity, World scietific, 2004. Preliminary version in Bulletin of the EATCS 77, pages 67–95, 2002.
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Goldreich, O., Krivelevich, M., Newman, I. et al. Hierarchy Theorems for Property Testing. comput. complex. 21, 129–192 (2012). https://doi.org/10.1007/s00037-011-0022-4
- Property testing
- hierarchy theorems
- query complexity
- graph properties
- monotone graph properties
- graph blow-up
- one-sided versus two-sided error
- adaptivity versus non-adaptivity