Log-rank and lifting for AND-functions
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Let f: {0,1}^n →{0, 1} be a boolean function, and let f_ (x, y) = f(x y) denote the AND-function of f, where x y denotes bit-wise AND. We study the deterministic communication complexity of f_ 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 f_. This comes within a log n factor of establishing the log-rank conjecturefor 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 𝔽_2-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 f_ 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 →{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 →ℝ with a larger range.
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