Let π : {0, 1} π β {0, 1} be a boolean function, and let π β§ (π₯, π¦) = π (π₯ β§π¦) denote the AND-function of π , where π₯ β§π¦ denotes bit-wise AND. We study the deterministic communication complexity of π β§ and show that, up to a log π factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of π β§ . This comes within a log π factor of establishing the log-rank conjecture for AND-functions with no assumptions on π . Our result stands in contrast with previous results on special cases of the logrank conjecture, which needed significant restrictions on π such as monotonicity or low F 2 -degree. Our techniques can also be used to prove (within a log π factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of π β§ is polynomially related to the AND-decision tree complexity of π .The results rely on a new structural result regarding boolean functions π : {0, 1} π β {0, 1} with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing π has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials π : {0, 1} π β R with a larger range. CCS CONCEPTSβ’ Theory of computation β Communication complexity; Oracles and decision trees.
Communication complexity studies the amount of communication necessary to compute a function whose value depends on information distributed among several entities. Yao [Yao79] initiated the study of communication complexity more than 40 years ago, and it has since become a central eld in theoretical computer science with many applications in diverse areas such as data structures, streaming algorithms, property testing, approximation algorithms, coding theory, and machine learning. The textbooks [KN06,RY20] provide excellent overviews of the theory and its applications.
Let f : {0, 1} n β {0, 1} be a boolean function, and let fβ§(x, y) = f (x β§ y) denote the AND-function of f , where x β§ y denotes bit-wise AND. We study the deterministic communication complexity of fβ§ and show that, up to a log n factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of fβ§. This comes within a log n factor of establishing the log-rank conjecture for AND-functions with no assumptions on f . Our result stands in contrast with previous results on special cases of the log-rank conjecture, which needed significant restrictions on f such as monotonicity or low F2-degree. Our techniques can also be used to prove (within a log n factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of fβ§ is polynomially related to the AND-decision tree complexity of f .The results rely on a new structural result regarding boolean functions f : {0, 1} n β {0, 1} with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing f has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials f : {0, 1} n β R with a larger range.
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