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Property Testing Lower Bounds via Communication Complexity

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Abstract

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds.

For the problem of testing whether a Boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010a). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

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Authors and Affiliations

  1. Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Eric Blais

  2. Computer Science Department, Aarhus University, Aarhus, Denmark

    Joshua Brody

  3. IIIS, Tsinghua University, Beijing, China

    Kevin Matulef

Authors
  1. Eric Blais
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  2. Joshua Brody
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  3. Kevin Matulef
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Correspondence to Joshua Brody.

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Blais, E., Brody, J. & Matulef, K. Property Testing Lower Bounds via Communication Complexity. comput. complex. 21, 311–358 (2012). https://doi.org/10.1007/s00037-012-0040-x

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  • DOI: https://doi.org/10.1007/s00037-012-0040-x

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