On Covering and Rank Problems for Boolean Matrices and Their Applications

Abstract

We consider combinatorial properties of Boolean matrices and their application to two-party communication complexity. Let A be a binary n x n matrix and let K be a field. Rectangles are sets of entries defined by collections of rows and columns. We denote by rankB(A) (rankK(A), resp.) the least size of a family of rectangles whose union (sum, resp.) equals A.We prove the following: - With probability approaching 1, for a random Boolean matrix A the following holds: rankb≥n(1−o(1)). - For finite K and fixed ε>O the following holds: If A is a Boolean matrix with rank B (A≤t) then there is some matrix \( A' \leqslant A \) such that \( A - A' \) has at most \( \varepsilon \cdot n^2 \) non-zero entries and rank K \( \left( {A'} \right) \leqslant t^{O\left( 1 \right)} \). As applications we mention some improvements of earlier results: (1) With probability approaching 1 a random n-variable Boolean function has nondeterministic communication complexity n, (2) functions with nondeterministic communication complexity l can be approximated by functions with parity communication complexity O(l). The latter complements a result saying that nondeterministic and parity communication protocols cannot efficiently simulate each other. Another consequence is: (3) matrices with small Boolean rank have small matrix rigidity over any field.KeywordsBoolean FunctionCommunication ComplexityRank ProblemMatrix RigidityBoolean MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.