A Study of the Binary and Boolean Rank of Matrices with Small Constant Real Rank
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arXiv
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| Format: | Preprint |
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2025
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| author | Parnas, Michal Shraibman, Adi |
| author_facet | Parnas, Michal Shraibman, Adi |
| contents | We initiate the study of the binary and Boolean rank of $0,1$ matrices that have a small rank over the reals. The relationship between these three rank functions is an important open question, and here we prove that when the real rank $d$ is a small constant, the gap between the real and the binary and Boolean rank is a small constant. We give tight upper and lower bounds on the Boolean and binary rank of matrices with real rank $1 \leq d \leq 4$, as well as determine the size of the largest isolation set in each case. Furthermore, we prove that for $d = 3,4$, the circulant matrix defined by a row with $d-1$ consecutive ones followed by $d-1$ zeros, is the only matrix of size $(2d-2)\times (2d-2)$ with real rank $d$ and Boolean and binary rank and isolation set of size $2d-2$, and this matrix achieves the maximal gap possible between the real and the binary and Boolean rank for these values of $d$.
Our results can also be interpreted in other equivalent terms, such as finding the minimal number of bicliques needed to partition or cover the edges of a bipartite graph whose reduced adjacency matrix has real rank $1 \leq d \leq 4$. We use a combination of combinatorial and algebraic techniques combined with the assistance of a computer program. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_05824 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Study of the Binary and Boolean Rank of Matrices with Small Constant Real Rank Parnas, Michal Shraibman, Adi Combinatorics We initiate the study of the binary and Boolean rank of $0,1$ matrices that have a small rank over the reals. The relationship between these three rank functions is an important open question, and here we prove that when the real rank $d$ is a small constant, the gap between the real and the binary and Boolean rank is a small constant. We give tight upper and lower bounds on the Boolean and binary rank of matrices with real rank $1 \leq d \leq 4$, as well as determine the size of the largest isolation set in each case. Furthermore, we prove that for $d = 3,4$, the circulant matrix defined by a row with $d-1$ consecutive ones followed by $d-1$ zeros, is the only matrix of size $(2d-2)\times (2d-2)$ with real rank $d$ and Boolean and binary rank and isolation set of size $2d-2$, and this matrix achieves the maximal gap possible between the real and the binary and Boolean rank for these values of $d$. Our results can also be interpreted in other equivalent terms, such as finding the minimal number of bicliques needed to partition or cover the edges of a bipartite graph whose reduced adjacency matrix has real rank $1 \leq d \leq 4$. We use a combination of combinatorial and algebraic techniques combined with the assistance of a computer program. |
| title | A Study of the Binary and Boolean Rank of Matrices with Small Constant Real Rank |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.05824 |