Binary Rank: Theory & Applications

• Binary rank is defined as the minimum number of rank-1 binary matrices (or monochromatic rectangles) whose disjoint supports partition the 1-entries of a 0-1 matrix.
• It distinguishes itself from related notions like Boolean rank and F2-linear rank by enforcing exact arithmetic and disjointness, impacting combinatorial and algebraic properties.
• Research on binary rank addresses NP-completeness, fixed-parameter algorithms, and unique augmentation phenomena, highlighting its role in matrix factorization and complexity theory.

Binary rank is a polysemous technical term. In the matrix literature, it commonly denotes the minimum number of rank-$1$ binary matrices whose supports partition the $1$-entries of a $0$-$1$ matrix; equivalently, it is the minimum number of monochromatic rectangles, or bicliques in the associated bipartite graph, needed to partition the support (Parnas, 20 Jan 2026). The same phrase is also used for the rank of an adjacency matrix over $\mathbb{F}_2$ in graph reduction theory (Pflueger, 2011), and for the Waring rank of a binary form in algebraic geometry (Moncusí et al., 2021). These usages share a binary ambient structure but encode different factorization models, invariants, and algorithmic questions.

1. Matrix binary rank as a combinatorial factorization invariant

For a binary matrix $M \in \{0,1\}^{m\times n}$, the matrix-theoretic binary rank $\operatorname{binrank}(M)$ is the minimal $d$ such that $M = A B$ over standard integer arithmetic with $A \in \{0,1\}^{m\times d}$, $1$0, and $1$1 entrywise (Parnas, 20 Jan 2026). Equivalently, $1$2 admits a decomposition

$1$3

into rank-$1$4 binary matrices with disjoint supports (Parnas, 20 Jan 2026). The disjointness condition is essential: it enforces that each $1$5-entry is produced exactly once, so binary rank is a partition parameter rather than merely a cover parameter.

This factorization has several equivalent formulations. Let $1$6 be the bipartite graph with biadjacency matrix $1$7. Then $1$8 equals the biclique partition number $1$9, and also equals $0$0, the minimum number of monochromatic combinatorial rectangles that partition the $0$1-entries of $0$2 (Parnas, 20 Jan 2026). These equivalences are the reason binary rank sits simultaneously in matrix factorization, extremal combinatorics, and communication complexity.

A closely related hierarchy is obtained by allowing controlled overlap. In the $0$3-binary-rank framework, $0$4 is the minimum number of monochromatic rectangles covering all $0$5-entries such that each $0$6-entry is covered by at most $0$7 rectangles; $0$8 recovers the binary rank, while $0$9 recovers the Boolean rank (Bshouty, 2023). This interpolation is useful both structurally and algorithmically, because it exposes which arguments depend on exact partitioning and which survive bounded overlap.

Notion	Defining operation	Equivalent view
Binary rank $1$0	$1$1 over standard arithmetic, $1$2, with no entry exceeding $1$3	Rectangle partition number; biclique partition number
Boolean rank $1$4	$1$5 under Boolean matrix multiplication	Rectangle cover number; biclique cover number
Rank over $1$6, $1$7	Linear rank over $1$8	Dimension of row or column space over $1$9

The distinction from $\mathbb{F}_2$0-rank is foundational. Over $\mathbb{F}_2$1, sums are taken modulo $\mathbb{F}_2$2; in binary rank, addition is ordinary arithmetic and the factorization must reproduce a $\mathbb{F}_2$3-$\mathbb{F}_2$4 matrix exactly. This structural difference is why binary rank matches rectangle partitions rather than linear independence over $\mathbb{F}_2$5 (Parnas et al., 2017).

2. Relations to Boolean rank, real rank, and lower-bound techniques

The basic inequalities for $\mathbb{F}_2$6-$\mathbb{F}_2$7 matrices are

$\mathbb{F}_2$8

where $\mathbb{F}_2$9 is the usual real rank and $M \in \{0,1\}^{m\times n}$0 is the Boolean rank (Parnas, 20 Jan 2026). The inequality $M \in \{0,1\}^{m\times n}$1 is immediate because a binary factorization is a special case of a real factorization; the inequality $M \in \{0,1\}^{m\times n}$2 reflects that a partition is, in particular, a cover (Parnas, 20 Jan 2026). By contrast, no general inequality holds between Boolean rank and real rank: the Boolean rank can be either larger or smaller depending on the matrix (Parnas et al., 2017).

Several structural properties of real rank fail for binary rank and Boolean rank. Real rank has the full-rank submatrix property, but binary and Boolean ranks do not: there are matrices of size $M \in \{0,1\}^{m\times n}$3 with binary rank $M \in \{0,1\}^{m\times n}$4 such that every $M \in \{0,1\}^{m\times n}$5 submatrix has binary rank at most $M \in \{0,1\}^{m\times n}$6, and there are matrices with Boolean rank $M \in \{0,1\}^{m\times n}$7 for which every proper submatrix has Boolean rank at most $M \in \{0,1\}^{m\times n}$8 (Parnas, 20 Jan 2026). This failure is one reason binary rank resists direct transplantation of linear-algebraic arguments.

Lower bounds often come from isolation sets. An isolation set is a set of $M \in \{0,1\}^{m\times n}$9-entries such that no two share a row or column and no two lie in the same $\operatorname{binrank}(M)$0 all-$\operatorname{binrank}(M)$1 submatrix. If $\operatorname{binrank}(M)$2 is the maximum isolation-set size, then

$\operatorname{binrank}(M)$3

and one also has the Dietzfelbinger–Hromkovič–Schnitger bound $\operatorname{binrank}(M)$4 (Parnas, 20 Jan 2026). These bounds are combinatorially elementary but frequently sharp on structured families.

The dependence of binary rank on matrix structure can be dramatic. The crown matrix $\operatorname{binrank}(M)$5 satisfies

$\operatorname{binrank}(M)$6

showing that binary and Boolean ranks can separate exponentially even when both are defined by rectangles on the same support pattern (Parnas, 20 Jan 2026). This is one of the canonical examples demonstrating that partition complexity and cover complexity are qualitatively different invariants.

3. Bases, augmentation, and non-linear-algebraic behavior

A central structural phenomenon is the failure of the augmentation property. For a rank function $\operatorname{binrank}(M)$7, a matrix $\operatorname{binrank}(M)$8 has the augmentation property if whenever each individual column $\operatorname{binrank}(M)$9 can be adjoined without increasing rank, then adjoining all of them simultaneously also does not increase rank. Real rank satisfies this trivially because linear span is closed, but binary rank and Boolean rank can fail badly (Parnas et al., 2017).

The exact characterization is given in terms of bases and the base graph. A base is a minimal spanning set of columns over the relevant semiring, and the base graph $d$0 has one vertex per base and a directed edge $d$1 if $d$2 spans $d$3 (Parnas et al., 2017). This graph is transitive and acyclic. The augmentation property for binary rank or Boolean rank holds if and only if there is a unique base that spans all other bases, equivalently, if and only if $d$4 has a single source (Parnas et al., 2017). This criterion replaces linear-span closure by a much more delicate partial-order structure on minimal generating sets.

For binary rank, the paper also provides concrete sufficient conditions. If an optimal binary decomposition $d$5 has the property that the rows of $d$6 are rows of $d$7 and $d$8 has the Unique base rows sums property, then the columns of $d$9 span every base of $M = A B$0, so the augmentation property holds (Parnas et al., 2017). A simpler sufficient condition is the existence of a disjoint-in-rows base together with the absence of identical rows (Parnas et al., 2017). These results show that uniqueness phenomena in binary factorizations are controlled by combinatorial restrictions on row dependencies rather than by ordinary linear independence.

The failure of augmentation can also be quantitatively extreme. For every $M = A B$1, there exists a matrix $M = A B$2 and columns $M = A B$3 such that each individual augmentation preserves binary rank,

$M = A B$4

but simultaneous augmentation yields

$M = A B$5

(Parnas et al., 2017). The Boolean-rank analogue also exhibits unbounded failure (Parnas et al., 2017). This places binary rank far from the behavior of classical linear rank.

The same paper uses augmentation to construct explicit separations from real rank. For every $M = A B$6, there exists a binary matrix $M = A B$7 with

$M = A B$8

and likewise a binary matrix with

$M = A B$9

(Parnas et al., 2017). These are linear, explicit gaps rather than existential asymptotics.

4. Complexity, parameterized algorithms, and property testing

Exact computation of Boolean rank and binary rank is NP-complete (Parnas, 20 Jan 2026). The corresponding approximation landscape is also hard: Boolean rank is NP-hard to approximate within $A \in \{0,1\}^{m\times d}$0 for $A \in \{0,1\}^{m\times d}$1 matrices (Parnas, 20 Jan 2026). Binary rank nevertheless admits fixed-parameter algorithms when parameterized by the target rank $A \in \{0,1\}^{m\times d}$2. A standard kernelization argument uses the fact that if $A \in \{0,1\}^{m\times d}$3 or $A \in \{0,1\}^{m\times d}$4, then $A \in \{0,1\}^{m\times d}$5 has at most $A \in \{0,1\}^{m\times d}$6 distinct rows and at most $A \in \{0,1\}^{m\times d}$7 distinct columns (Parnas, 20 Jan 2026). This yields kernels of size at most $A \in \{0,1\}^{m\times d}$8, after which exhaustive search becomes possible, although the dependence on $A \in \{0,1\}^{m\times d}$9 remains double-exponential or worse (Parnas, 20 Jan 2026).

The same distinct-row and distinct-column phenomenon underlies property testing. A one-sided adaptive tester for binary rank at most $1$00 with query complexity $1$01, and a non-adaptive one with $1$02, were given in work on testing Boolean and binary rank (Parnas et al., 2019). For Boolean rank, the same paper gave a one-sided non-adaptive tester with query complexity $1$03 (Parnas et al., 2019). The binary-rank testers grow a sampled submatrix until the number of distinct rows or columns exceeds $1$04, at which point low binary rank becomes impossible (Parnas et al., 2019).

The $1$05-binary-rank framework sharpens this picture. If $1$06, then the number of distinct rows and columns satisfies

$1$07

and this bound is tight (Bshouty, 2023). Using this combinatorial bound, one obtains one-sided adaptive and non-adaptive testers for $1$08-binary rank at most $1$09 with query complexities

$1$10

respectively (Bshouty, 2023). For fixed $1$11, this improves the earlier tester by a factor of $1$12 (Bshouty, 2023).

These testing results are conceptually significant because they bypass the hardness of exact computation. They also reinforce the role of rectangle structure: the testers do not search for optimal partitions directly, but instead detect the combinatorial signatures that all low-rank partitions must impose on submatrices.

5. Exact values, extremal families, and small-real-rank regimes

A substantial body of work studies binary rank on structured matrix families. For matrices of small constant real rank, tight bounds are known for $1$13 (Parnas et al., 8 Jul 2025).

Real rank $1$14	Binary rank bounds	Boolean rank bounds
$1$15	$1$16	$1$17
$1$18	$1$19	$1$20
$1$21	$1$22	$1$23
$1$24	$1$25	$1$26

For $1$27, the extremal matrices are circulants. The circulant $1$28, formed from a first row with $1$29 consecutive ones followed by $1$30 zeros, satisfies

$1$31

and for $1$32 it is the unique matrix of size $1$33 with this extremal behavior (Parnas et al., 8 Jul 2025). This identifies the sharp maximum gap possible for these real-rank values.

Regular matrices yield a different kind of extremality. For infinitely many $1$34, there exists a square regular $1$35-$1$36 matrix $1$37 such that

$1$38

where $1$39 is the complement (Haviv et al., 2022). Equivalently, the ones in $1$40 can be partitioned into $1$41 rectangles, while covering the zeros of $1$42 requires quasi-polynomially more rectangles (Haviv et al., 2022). This settles, in the regular setting, longstanding questions about the possible disparity between a matrix and its complement.

Circulant block-diagonal matrices admit especially explicit formulas. If $1$43 is the complement of a $1$44 circulant block-diagonal matrix with $1$45, then

$1$46

(Haviv et al., 2022). The same paper proves lower bounds such as

$1$47

for the single-block circulant $1$48, significantly generalizing Gregory’s result $1$49 (Haviv et al., 2022). For $1$50-regular matrices, it also determines exactly the smallest possible binary rank of the complement as a function of the original binary rank: $1$51 among all $1$52-regular $1$53 with $1$54 (Haviv et al., 2022).

6. Other established meanings of “binary rank”

In graph reduction theory, the binary rank of a graph $1$55 is the rank of its adjacency matrix over the field with two elements: $1$56 If $1$57 has $1$58 vertices, its nullity is $1$59 (Pflueger, 2011). This invariant governs reduction dynamics arising in gene assembly: for any complete reduction from $1$60 to the empty graph, the number of negative-rule applications is exactly $1$61 (Pflueger, 2011). The theory also gives a reduction formula

$1$62

for reducible vertex sets $1$63, establishes path invariance of reductions, and relates reductions to principal pivot transforms (Pflueger, 2011). Here “binary rank” is purely $1$64-linear and has no rectangle-partition interpretation.

In algebraic geometry and tensor decomposition, binary rank refers to the Waring rank of a binary form. For a degree-$1$65 binary form $1$66, it is the minimal $1$67 such that

$1$68

(Moncusí et al., 2021). For binary binomials, there is an explicit case-by-case formula in terms of the parameters $1$69 and $1$70 after reducing to the normal form $1$71 (Moncusí et al., 2021). Over $1$72, a degree-$1$73 binary form $1$74 with distinct roots has maximal real rank $1$75 if and only if it has $1$76 real roots, equivalently if and only if every directional derivative $1$77 has $1$78 real roots (Causa et al., 2010). At the level of typical real ranks, the algebraic boundaries between typical-rank regions are unions of duals to suitable coincident root loci (Brambilla et al., 2019). In this context, “binary” refers to the number of variables, not to $1$79-$1$80 combinatorics.

Taken together, these usages show that binary rank is not a single invariant but a family of distinct notions tied to different binary structures: $1$81-$1$82 matrix partitions, $1$83-linear graph theory, and symmetric tensor decomposition of binary forms. Any technical discussion therefore requires the ambient category to be fixed explicitly.