Upper Bounds on the Boolean Rank of Kronecker Products

The Boolean rank of a $0,1$-matrix $A$, denoted $R_\mathbb{B}(A)$, is the smallest number of monochromatic combinatorial rectangles needed to cover the $1$-entries of $A$. In 1988, de Caen, Gregory, and Pullman asked if the Boolean rank of the Kronecker product $C_n \otimes C_n$ is strictly smaller than the square of $R_\mathbb{B}(C_n)$, where $C_n$ is the $n \times n$ matrix with zeros on the diagonal and ones everywhere else (Carib. Conf. Comb. & Comp., 1988). A positive answer was given by Watts for $n=4$ (Linear Alg. and its Appl., 2001). A result of Karchmer, Kushilevitz, and Nisan, motivated by direct-sum questions in non-deterministic communication complexity, implies that the Boolean rank of $C_n \otimes C_n$ grows linearly in that of $C_n$ (SIAM J. Disc. Math., 1995), and thus $R_\mathbb{B}(C_n \otimes C_n) < R_\mathbb{B}(C_n)^2$ for every sufficiently large $n$. Their proof relies on a probabilistic argument. In this work, we present a general method for proving upper bounds on the Boolean rank of Kronecker products of $0,1$-matrices. We use it to affirmatively settle the question of de Caen et al. for all integers $n \geq 7$. We further provide an explicit construction of a cover of $C_n \otimes C_n$, whose number of rectangles nearly matches the optimal asymptotic bound. Our method for proving upper bounds on the Boolean rank of Kronecker products might find applications in different settings as well. We express its potential applicability by extending it to the wider framework of spanoids, recently introduced by Dvir, Gopi, Gu, and Wigderson (SIAM J. Comput., 2020).