Mathematical and computational perspectives on the Boolean and binary rank and their relation to the real rank

Abstract: This survey provides a comprehensive overview of the study of the binary and Boolean rank from both a mathematical and a computational perspective, with particular emphasis on their relationship to the real rank. We review the basic definitions of these rank functions and present the main alternative formulations of the binary and Boolean rank, together with their computational complexity and their deep connection to the field of communication complexity. We summarize key techniques used to establish lower and upper bounds on the binary and Boolean rank, including methods from linear algebra, combinatorics and graph theory, isolation sets, the probabilistic method, kernelization, communication protocols and the query to communication lifting technique. Furthermore, we highlight the main mathematical properties of these ranks in comparison with those of the real rank, and discuss several non-trivial bounds on the rank of specific families of matrices. Finally, we present algorithmic approaches for computing and approximating these rank functions, such as parameterized algorithms, approximation algorithms, property testing and approximate Boolean matrix factorization (BMF). Together, the results presented outline the current theoretical knowledge in this area and suggest directions for further research.