Log-rank and lifting for AND-functions

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 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 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.