The Partition Bound for Classical Communication Complexity and Query Complexity

Abstract

In this work we introduce, both for classical communication complexity and query complexity, a modification of the partition bound introduced by Jain and Klauck [JK10]. We call it the public-coin partition bound. We show that (the logarithm to the base two of) its communication complexity and query complexity versions form, for all relations, a quadratically tight lower bound on the public-coin randomized communication complexity and randomized query complexity respectively.The partition bound introduced by Jain and Klauck [JK10] is known to be one of the strongest lower bound methods in classical communication complexity and query complexity. It is known to be stronger than most other lower bound methods, both in communication complexity and query complexity, except its relationship with the information complexity lower bound method in communication complexity is unknown. It is an interesting open question, in both these settings, as to how tight this lower bound method is. We are not aware, to the best of our knowledge, of any function or relation where this method is asymptotically weaker either for communication complexity or for query complexity.In this work we introduce, both for communication complexity and query complexity, a modification of the partition bound which we call the public-coin partition bound. Analogous to the partition bound, our new bound is also a linear-programming based lower bound method. We show that (the logarithm to the base two of) its communication and query complexity versions continue to form a lower bound on the public-coin communication complexity and randomized query complexity respectively. In addition we show that the square of (the logarithm to the base two of) its communication and query complexity versions form an upper bound on the public-coin communication complexity and randomized query complexity respectively. Also it is easily seen via their linear programs that our new bound is stronger than the partition bound for all relations, both in communication complexity and query complexity.

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“…The definition of the corruption bound Corr( f ) is given in Section 3, but for now, we note that Corr( f ) essentially depends on the size of the largest approximately 1-monochromatic rectangle in the communication matrix of f . (For an extensive discussion of the different lower bound methods in communication complexity, see [24].) Previously, Klauck [29] showed that Corr( f ) lies somewhere between the MA and AM communication complexities of f ; namely…”

“…A lot of effort (e. g., [29,24,28,25,19]) has been spent on comparing the relative strengths of different lower bound methods in communication complexity with the goal of finding a natural method that captures the bounded-error randomized communication complexity of every function. Theorem 1.1 can be viewed as achieving a diametrically opposite goal: we THEORY OF COMPUTING, Volume 12 (9), 2016, pp.…”

Interactive Information Complexity

“…recently, the lower bounds of [47] have been shown [27] to not apply to the partition bound technique of Jain and Klauck [35].…”

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“…Settling this claim requires the introduction of a second measure, call it λ , to account for covering by large rectangles. The smooth rectangle bound [9] was discovered very recently and overcomes limitations of Yao's corruption bound-at the expense of being more challenging to use.…”

“…The authors actually derive a much stronger statement, giving a detailed characterization of the distribution of x, y . To complete the proof, they use a criterion for high communication complexity due to Jain and Klauck [9], known as the smooth rectangle bound. Specifically, Chakrabarti and Regev use their anticoncentration result to argue that in any partition of R n × R n , only a small constant measure of inputs can be covered by large uncorrupted rectangles.…”