An additive combinatorics approach to the log-rank conjecture in communication complexity

For a ${0,1}$-valued matrix $M$ let $\rm{CC}(M)$ denote the deterministic communication complexity of the boolean function associated with $M$. The log-rank conjecture of Lov'{a}sz and Saks [FOCS 1988] states that $\rm{CC}(M) \leq \log^c(\rm{rank}(M))$ for some absolute constant $c$ where $\rm{rank}(M)$ denotes the rank of $M$ over the field of real numbers. We show that $\rm{CC}(M)\leq c \cdot \rm{rank}(M)/\log \rm{rank}(M)$ 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].

💡 Research Summary

The paper tackles one of the most enduring open problems in communication complexity: the log‑rank conjecture, which posits that the deterministic communication complexity CC(M) of a Boolean matrix M should be bounded by a polylogarithmic function of its real‑rank, i.e., CC(M)=O(log rank(M)). While the best known upper bound to date is linear (CC(M)≤0.415·rank(M) due to Kotlov) and the lower bound is only Ω(log rank(M)), the authors obtain a substantially stronger bound under a well‑studied hypothesis from additive combinatorics, the Polynomial Freiman‑Ruzsa (PFR) conjecture.

The core of the argument is a refined analysis of “approximate duality”, a notion introduced by Ben‑Sasson and Zewi (STOC 2011). For subsets A,B⊆F₂ⁿ the duality measure D(A,B) is defined as the expectation of (−1)^{⟨a,b⟩} over uniformly random a∈A and b∈B. When D(A,B)=1, A lies inside an affine shift of the orthogonal complement of B, i.e., the pair is perfectly dual. Earlier work showed that if D(A,B) is very close to 1, one can extract large sub‑sets A′⊆A, B′⊆B with D(A′,B′)=1. Moreover, assuming the PFR conjecture, a similar statement holds even when D(A,B) is exponentially small.

The authors improve this line of work in two significant ways. First, they prove a “main technical lemma” (Lemma 1.10) stating that if D(A,B)≥2^{−√n} then, assuming PFR, there exist sub‑sets A′,B′ of size at least 2^{−c n/ log n}·|A| and 2^{−c n/ log n}·|B| respectively such that D(A′,B′)=1. This improves the earlier bound of 2^{−δ n} (for a constant δ) both in exponent and in independence from the original set sizes. Second, they introduce the concept of the α‑spectrum of a set B, namely the set of vectors x for which the bias |E_{b∈B}