An Additive Combinatorics Approach Relating Rank to Communication Complexity

Abstract

For a {0, 1}-valued matrix M let CC(M ) denote the deterministic communication complexity of the boolean function associated with M . The log-rank conjecture of Lovász and Saks [FOCS 1988] states that CC(M ) ≤ log c (rank(M )) for some absolute constant c where rank(M ) denotes the rank of M over the field of real numbers. We show that CC(M ) ≤ c · rank(M )/ log rank(M ) for some absolute constant c, assuming a well-known conjecture from additive combinatorics known as the Polynomial Freiman-Ruzsa (PFR) conjecture.Our proof is based on the study of the "approximate duality conjecture" which was recently suggested by Ben-Sasson and Zewi [STOC 2011] and studied there in connection to the PFR conjecture. First we improve the bounds on approximate duality assuming the PFR conjecture. Then we use the approximate duality conjecture (with improved bounds) to get the aforementioned upper bound on the communication complexity of low-rank martices, where this part uses the methodology suggested by Nisan and Wigderson [Combinatorica 1995].

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“…Now, the main observation is that the number of distinct inner product patterns ( a i , b ) i∈ [d] for b ∈ B 0 is bounded by A serious shortcoming of Theorem 4.29 is that it applies only to sets which are already nearly orthogonal, and moreover, it loses an exponential factor. The following result of Ben-Sasson, Lovett and Ron-Zewi [10] improves both of these. However, it assumes the Polynomial Freiman-Ruzsa conjecture.…”

“…It was introduced by Ben-Sasson and RonZewi [11] who used it to show that a family of potential constructions of bipartite Ramsey graphs must also be two-source extractors. Later works used the notion of approximate duality to give an improvement on the known upper bounds for the log-rank conjecture in communication complexity [10] and to prove lower bounds on certain families of locally decodable codes [12]. Let A, B ⊂ F n be two subsets of a vector space.…”

“…The best result to date appears in [10] where a weak version of the approximate duality conjecture in F n 2 is proved, assuming the Polynomial Freiman-Ruzsa conjecture in F n 2 . The result was generalized to groups of constant torsion in [12] where it was used to prove lower bounds for certain families of locally decodable codes.…”

On the structure of the spectrum of small sets

“…Our interest in this problem originates from applications of it in computational complexity, where a better understanding of the structure of the spectrum of small sets can help to shed light on some of the main open problems in the area, such as constructions of two source extractors [1,2,3] or the log rank conjecture in communication complexity [4]. We refer the interested reader to a survey by the second author on applications of additive combinatorics in theoretical computer science [5].…”

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“…However, currently it seems to be insufficient for the other applications discussed above [18,3,4], mainly because they require structural result for very large doubling constants. As mentioned, the result of Aggarwal et al [1] already uses Sanders' theorem.…”