Mathematical and computational perspectives on the Boolean and binary rank and their relation to the real rank
Michal Parnas
TL;DR
The survey addresses how the Boolean and binary ranks of 0-1 matrices relate to the real rank, clarifying that $rank_{\mathbb{R}}(M) \leq rank_{bin}(M)$ and $rank_{\mathbb{B}}(M) \leq rank_{bin}(M)$ with complex, non-monotone gaps between $rank_{\mathbb{B}}(M)$ and $rank_{\mathbb{R}}(M)$. It consolidates multiple equivalent formulations—monochromatic rectangle coverings, biclique covers in bipartite graphs, intersection numbers, and set basis problems—showing how these viewpoints yield complementary tools from linear algebra, combinatorics, and graph theory. The paper surveys computational complexity results, noting NP-hardness of computing Boolean and binary ranks, and discusses lower/upper bound techniques, including lifting, kernelization, and communication protocols, as well as algorithmic approaches like parameterized and approximation methods. By connecting these ranks to communication complexity and diverse applications, the survey provides a unified framework to transfer insights across fields and identifies directions for future work in tight bounds and efficient algorithms. The work highlights the rich interplay between algebraic, combinatorial, and computational perspectives in understanding rank functions beyond 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.