Property Testing Lower Bounds via Communication Complexity

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 [25]. 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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“…Recently, a very elegant lower bound method for classical property testing was developed by Blais et al [38], based on communication complexity. In the basic setting of communication complexity [172,115], two parties (Alice with input x and Bob with input y) try to compute a function F(x, y) that depends on both of their inputs, using as little communication as possible.…”

“…Blais et al [38] showed for the first time how to derive property testing lower bounds from communication complexity. Their idea is to convert a T -query property tester for some property P into a protocol for some related communication problem F, by showing that 1-inputs (x, y) for F somehow correspond to elements of P, while 0-inputs (x, y) for F correspond to elements that are far from P. The more efficient the tester, the less communication the protocol needs.…”

“…Can some modification of the ideas of Blais et al [38] be used to obtain non-trivial lower bounds on quantum testers?…”

“…We also describe the adversary method, which-when applied properly-proves optimal lower bounds. And we ask whether the recent classical property testing lower bounds of Blais et al [38], based on communication complexity, can be applied to quantum testers as well.…”

Practical verified computation with streaming interactive proofs

“…More recent motivations include connections to property testing [17]. A clever protocol of Håstad and Wigderson [63] gives an optimal O(m) communication protocol for m-disj, improving upon the trivial O(m log n) and the easy O(m log m) bounds.…”