Log-rank and lifting for AND-functions

Abstract

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 logrank 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. CCS CONCEPTS• Theory of computation → Communication complexity; Oracles and decision trees.

A Note on the LogRank Conjecture in Communication Complexity

“…The 35-year-old conjecture is widely open today, inspiring numerous theorems and approaches even in the last few years [21][22][23][24][25][26][27].…”

Dimension-free bounds and structural results in communication complexity

“…We conjecture that the exponential equivalence between D(F ∧ ) and dt ∧ (f ) in Proposition 6.1 can be improved to a polynomial equivalence. Recently, [KLMY20] proved dt ∧ (f ) = O(D(f ∧ ) 3 log n), but due to the log(n) factor, their statement comes short of establishing this conjecture. Now, let us turn to randomized communication complexity and its related matrix parameters such as the trace and the γ 2 norm.…”

Algebraic Representations of Unique Bipartite Perfect Matching

“…It is well known [KLMY21,BC99] that the rank of the communication matrices M f∧ and M f⊕ is exactly characterized by the sparsity (i.e., number of monomials) of the polynomials representing f in the 0, 1basis and the {±1}-basis (the Fourier basis), respectively. In other words,…”