Boolean Rank Research Articles

On the binary and Boolean rank of regular matrices

A 0,1 matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers k, there exists a square regular 0,1 matrix with binary rank k, such that the Boolean rank of its complement is kΩ˜(log⁡k). This settles, in a strong form, a question of Pullman (1988) [27] and a conjecture of Hefner et al. (1990) [18]. The result can be viewed as a regular analogue of a recent result of Balodis et al. (2021) [2], motivated by the clique vs. independent set problem in communication complexity and by the (disproved) Alon-Saks-Seymour conjecture in graph theory. As an application of the produced regular matrices, we obtain regular counterexamples to the Alon-Saks-Seymour conjecture and prove that for infinitely many integers k, there exists a regular graph with biclique partition number k and chromatic number kΩ˜(log⁡k).

Upper bounds on the Boolean rank of Kronecker products

The Boolean rank of a 0,1-matrix A , denoted R B ( A ) , is the smallest number of monochromatic combinatorial rectangles needed to cover the 1-entries of A . In 1988, de Caen, Gregory, and Pullman asked if the Boolean rank of the Kronecker product C n ⊗ C n is strictly smaller than the square of R B ( C n ) , where C n is the n × n matrix with zeros on the main diagonal and ones everywhere else (de Caen et al., 1988). A positive answer was given by Watts for n = 4 (Watts, 2001). A result of Karchmer, Kushilevitz, and Nisan, motivated by direct-sum questions in non-deterministic communication complexity, implies that the Boolean rank of C n ⊗ C n grows linearly in that of C n (Karchmer et al., 1995), and thus R B ( C n ⊗ C n ) < R B ( C n ) 2 for every sufficiently large n . Their proof relies on a probabilistic argument. In this work, we present a general method for proving upper bounds on the Boolean rank of Kronecker products of 0,1-matrices. We use it to affirmatively settle the question of de Caen et al. for all integers n ≥ 7 . We further provide an explicit construction of a cover of C n ⊗ C n , whose number of rectangles nearly matches the optimal asymptotic bound. Our method for proving upper bounds on the Boolean rank of Kronecker products might find applications in different settings as well. We express its potential applicability by extending it to the wider framework of spanoids , recently introduced by Dvir et al. (2020).

Property Testing of the Boolean and Binary Rank

We present algorithms for testing if a (0,1)-matrix M has Boolean/binary rank at most d, or is 𝜖-far from having Boolean/binary rank at most d (i.e., at least an 𝜖-fraction of the entries in M must be modified so that it has rank at most d). For the Boolean rank we present a non-adaptive testing algorithm whose query complexity is \(\tilde {O}\left (d^{4}/ \epsilon ^{6}\right )\). For the binary rank we present a non-adaptive testing algorithm whose query complexity is O(22d/𝜖2), and an adaptive testing algorithm whose query complexity is O(22d/𝜖). All algorithms are 1-sided error algorithms that always accept M if it has Boolean/binary rank at most d, and reject with probability at least 2/3 if M is 𝜖-far from having Boolean/binary rank at most d.

Linear Operators That Preserve Arctic Ranks of Boolean Matrices

We study some properties of arctic rank of Boolean matrices. We compare the arctic rank with Boolean rank and term rank of a given Boolean matrix. Furthermore, we obtain some characterizations of linear operators that preserve arctic rank on Boolean matrix space.

Upper Bounds on the Boolean Rank of Kronecker Products

The Boolean rank of a 0,1-matrix A, denoted Rb(A), is the smallest number of monochromatic combinatorial rectangles needed to cover the 1-entries of A. In 1988, de Caen, Gregory, and Pullman asked if the Boolean rank of the Kronecker product Cn ® Cn is strictly smaller than the square of RB(Cn), where Cn is the n x n matrix with zeros on the diagonal and ones everywhere else (Carib. Conf. Comb. & Comp., 1988). A positive answer was given by Watts for n = 4 (Linear Alg. and its Appl., 2001). A result of Karchmer, Kushilevitz, and Nisan, motivated by direct-sum questions in non-deterministic communication complexity, implies that the Boolean rank of Cn ® Cn grows linearly in that of Cn (SIAM J. Disc. Math., 1995), and thus Rb(Cn ® Cn) < RB(Cn)2 for every sufficiently large n. Their proof relies on a probabilistic argument.In this work, we present a general method for proving upper bounds on the Boolean rank of Kronecker products of 0,1-matrices. We use it to affirmatively settle the question of de Caen et al. for all integers n > 7. We further provide an explicit construction of a cover of Cn ® Cn, whose number of rectangles nearly matches the optimal asymptotic bound. Our method for proving upper bounds on the Boolean rank of Kronecker products might find applications in different settings as well. We express its potential applicability by extending it to the wider framework of spanoids, recently introduced by Dvir, Gopi, Gu, and Wigderson (SIAM J. Comput., 2020).

Minimum-rank and maximum-nullity of graphs and their linear preservers

ABSTRACT The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose entry (for ) is nonzero whenever is an edge in G and is zero otherwise. The sum of the minimum rank of a graph and its maximum nullity (similarly defined) is always the number of vertices in G. This article compares the minimum rank with the clique covering number of G and the Boolean rank of its adjacency matrix. It does the same analysis for bipartite graphs. Finally, we investigate the linear operators on the set of graphs on n vertices that preserve the minimum rank.

Parameterized low-rank binary matrix approximation

Low-rank binary matrix approximation is a generic problem where one seeks a good approximation of a binary matrix by another binary matrix with some specific properties. A good approximation means that the difference between the two matrices in some matrix norm is small. The properties of the approximation binary matrix could be: a small number of different columns, a small binary rank or a small Boolean rank. Unfortunately, most variants of these problems are NP-hard. Due to this, we initiate the systematic algorithmic study of low-rank binary matrix approximation from the perspective of parameterized complexity. We show in which cases and under what conditions the problem is fixed-parameter tractable, admits a polynomial kernel and can be solved in parameterized subexponential time.

The Boolean rank of the uniform intersection matrix and a family of its submatrices

We study the Boolean rank of two families of binary matrices. The first is the binary matrix Ak,t that represents the adjacency matrix of the intersection bipartite graph of all subsets of size t of {1,2,...,k}. We prove that its Boolean rank is k for every k≥2t.The second family is the family Us,m of submatrices of Ak,t that is defined as Us,m=(Jm⊗Is)+(I¯m⊗Js), where Is is the identity matrix, Js is the all-ones matrix, s=k−2t+2 and m=(2t−2t−1). We prove that the Boolean rank of Us,m is also k for the following values of t and s: for s=2 and any t≥2, that is k=2t; for t=3 and any s≥2; and for any t≥2 and s>2t−2, that is k>4t−4.

The augmentation property of binary matrices for the binary and Boolean rank

We define the augmentation property for binary matrices with respect to different rank functions. A matrix A has the augmentation property for a given rank function, if for any subset of column vectors x1,...,xt for which the rank of A does not increase when augmented separately with each of the vectors xi, 1≤i≤t, it also holds that the rank does not increase when augmenting A with all vectors x1,...,xt simultaneously. This property holds trivially for the usual linear rank over the reals, but as we show, things change significantly when considering the binary and Boolean rank of a matrix.We prove a necessary and sufficient condition for this property to hold under the binary and Boolean rank of binary matrices. Namely, a matrix has the augmentation property for these rank functions if and only if it has a unique base that spans all other bases of the matrix with respect to the given rank function. For the binary rank, we also present a concrete sufficient characterization of a family of matrices that has the augmentation property. This characterization is based on the possible types of linear dependencies between rows of V, in optimal binary decompositions of the matrix as A=U⋅V.

English

Let ℤ+ be the semiring of all nonnegative integers and A an m × n matrix over ℤ+. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m×n matrices whose isolation number is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.

On the linear extension complexity of regular n-gons

In this paper, we propose new lower and upper bounds on the linear extension complexity of regular n-gons. Our bounds are based on the equivalence between the computation of (i) an extended formulation of size r of a polytope P, and (ii) a rank-r nonnegative factorization of a slack matrix of the polytope P. The lower bound is based on an improved bound for the rectangle covering number (also known as the boolean rank) of the slack matrix of the n-gons. The upper bound is a slight improvement of the result of Fiorini, Rothvoss and Tiwary (2012) [9]. The difference with their result is twofold: (i) our proof uses a purely algebraic argument while Fiorini et al. used a geometric argument, and (ii) we improve the base case allowing us to reduce their upper bound 2⌈log2⁡(n)⌉ by one when 2k−1<n≤2k−1+2k−2 for some integer k. We conjecture that this new upper bound is tight, which is suggested by numerical experiments for small n. Moreover, this improved upper bound allows us to close the gap with the best known lower bound for certain regular n-gons (namely, 9≤n≤13 and 21≤n≤24) hence allowing for the first time to determine their extension complexity.

On Ranks of Regular Polygons

In this paper we study various versions of extension complexity for polygons through the study of factorization ranks of their slack matrices. In particular, we develop a new asymptotic lower bound for their nonnegative rank, decreasing the gap between the current bounds, we introduce a new upper bound for their boolean rank, deriving some numerical evidence for the asymptotic equivalence of the boolean rank of $n$-gons and $\log_2(n)$, and we show the nonmonotonicity of the complex positive semidefinite rank of $n$-gons.

Exact and approximate Boolean matrix decomposition with column-use condition

An arbitrary $$m\times n$$ Boolean matrix M can be decomposed exactly as $$M =U\circ V$$ , where U (resp. V) is an $$m\times k$$ (resp. $$k\times n$$ ) Boolean matrix and $$\circ $$ denotes the Boolean matrix multiplication operator. The minimum k is called the Boolean rank of M, and it is known to be NP-hard to find it. With the interpretability issue in data mining applications in mind, we impose the column-use condition that the columns of U form a subset of the columns of the given M, and employ commonly used heuristics to find as small a k as possible.To this end, we first derive an exact closed-form formula, $$J=\overline{\overline{M}^\mathrm{T}\circ M}$$ , such that $$M =M\circ J^\mathrm{T}$$ holds, where J is maximal in the sense that if any 0 element in J is changed to a 1; then, this equality no longer holds. We measure the performance (in minimizing k) of our algorithms on several real benchmark datasets. The results demonstrate that one of our proposed algorithms performs as well or better on all but one of them than other representative heuristic algorithms, which do not impose the column-use condition and thus theoretically should find a smaller k.Boolean matrix decomposition with the column-use condition has wide applications. In educational databases, for example, the “ideal item response matrix” R, the “knowledge state matrix” A, and the “Q-matrix” Q play important roles. As they are related exactly by $$\overline{R}=\overline{A}\circ Q^\mathrm{T}$$ , given R, we can find A and Q with a small number (k) of interpretable “knowledge states,” using our heuristics.

LINEAR PRESERVERS OF BOOLEAN RANK BETWEEN DIFFERENT MATRIX SPACES

The Boolean rank of a nonzero m × n Boolean matrix A is the least integer k such that there are an m × k Boolean matrix B and a k × n Boolean matrix C with A = BC. We investigate the structure of linear transformations T : Mm,n ! Mp,q which preserve Boolean rank. We also show that if a linear transformation preserves the set of Boolean rank 1 matrices and the set of Boolean rank k matrices for any k, 2 � k � min{m,n} (or if T strongly preserves the set of Boolean rank 1 matrices), then T preserves all Boolean ranks.

Tropical lower bounds for extended formulations

The tropical arithmetic operations on $$\mathbb {R}$$R, defined as $$\oplus :(a,b)\rightarrow \min \{a,b\}$$?:(a,b)?min{a,b} and $$\otimes :(a,b)\rightarrow a+b$$?:(a,b)?a+b, arise from studying the geometry over non-Archimedean fields. We present an application of tropical methods to the study of extended formulations for convex polytopes. We propose a non-Archimedean generalization of the well known Boolean rank bound for the extension complexity. We show how to construct a real polytope with the same extension complexity and combinatorial type as a given non-Archimedean polytope. Our results allow us to develop a method of constructing real polytopes with large extension complexity.

Linear operators that preserve graphical properties of matrices: Isolation numbers

Let A be a Boolean {0, 1} matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A. A linear operator on the set of m × n Boolean matrices is a mapping which is additive and maps the zero matrix, O, to itself. A mapping strongly preserves a set, S, if it maps the set S into the set S and the complement of the set S into the complement of the set S. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that T is a Boolean linear operator that strongly preserves isolation number k for any 1 ⩽ k ⩽ min{m, n} if and only if there are fixed permutation matrices P and Q such that for \(X \in \mathcal{M}_{m,n} (\mathbb{B})T(X) = PXQ\) or, m = n and T (X) = PXtQ where Xt is the transpose of X.

CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS OVER BOOLEAN MATRICES

The Boolean rank of a nonzero m <TEX>$m{\times}n$</TEX> Boolean matrix A is the least integer k such that there are an <TEX>$m{\times}k$</TEX> Boolean matrix B and a <TEX>$k{\times}n$</TEX> Boolean matrix C with A = BC. In 1984, Beasley and Pullman showed that a linear operator preserves the Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks 1 and 2. In this paper, we extend this characterization of linear operators that preserve the Boolean ranks of Boolean matrices. We show that a linear operator preserves all Boolean ranks if and only if it preserves two consecutive Boolean ranks if and only if it strongly preserves a Boolean rank k with <TEX>$1{\leq}k{\leq}min\{m,n\}$</TEX>.

The primitive Boolean matrices with the second largest scrambling index by Boolean rank

The scrambling index of an n × n primitive Boolean matrix A is the smallest positive integer k such that Ak(AT)k = J, where AT denotes the transpose of A and J denotes the n×n all ones matrix. For an m×n Boolean matrix M, its Boolean rank b(M) is the smallest positive integer b such that M = AB for some m × b Boolean matrix A and b×n Boolean matrix B. In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an n×n primitive matrix M in terms of its Boolean rank b(M), and they also characterized all primitive matrices that achieve the upper bound. In this paper, we characterize primitive Boolean matrices that achieve the second largest scrambling index in terms of their Boolean rank.

THE COMPETITION INDEX OF A NEARLY REDUCIBLE BOOLEAN MATRIX

Cho and Kim [4] have introduced the concept of the competition index of a digraph. Similarly, the competition index of an <TEX>$n{\times}n$</TEX> Boolean matrix A is the smallest positive integer q such that <TEX>$A^{q+i}(A^T)^{q+i}=A^{q+r+i}(A^T)^{q+r+i}$</TEX> for some positive integer r and every nonnegative integer i, where <TEX>$A^T$</TEX> denotes the transpose of A. In this paper, we study the upper bound of the competition index of a Boolean matrix. Using the concept of Boolean rank, we determine the upper bound of the competition index of a nearly reducible Boolean matrix.

Linear operators that preserve Boolean rank of Boolean matrices

The Boolean rank of a nonzero m × n Boolean matrix A is the minimum number k such that there exist an m× k Boolean matrix B and a k × n Boolean matrix C such that A = BC. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and 2. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and k for some 1 < k ⩽ m.