Related papers: Most binary matrices have no small defining set

If $D$ is a partially filled-in $(0,1)$-matrix with a unique completion to a $(0,1)$-matrix $M$ (with prescribed row and column sums), we say that $D$ is a {\em defining set} for $M$. A {\em critical set} is a minimal defining set (the…

A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial…

Let $M$ be a $n\times m$ $(0,1)$-matrix. We define the $s$-binary rank, $br_s(M)$, of $M$ to be the minimal integer $d$ such that there are $d$ monochromatic rectangles that cover all the $1$-entries in the matrix, and each $1$-entry is…

We show that, for each real number $\epsilon > 0$ there is an integer $c$ such that, if $M$ is a simple triangle-free binary matroid with $|M| \ge (\tfrac{1}{4} + \epsilon) 2^{r(M)}$, then $M$ has critical number at most $c$. We also give a…

This work shows that the smallest natural number $d_n$ that is not the determinant of some $n\times n$ binary matrix is at least $c\,2^n/n$ for $c=1/201$. That same quantity naturally lower bounds the number of distinct integers $D_n$ which…

Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial…

A defining set of a Latin square is a partially filled-in Latin square which completes to no other Latin square of the same order. We introduce the concept of a $k$-strong defining set, in which if less than $k$ entries are deleted, the…

Consider the $2n$-by-$2n$ matrix $M=(m_{i,j})_{i,j=1}^{2n}$ with $m_{i,j} = 1$ for $i,j$ satisfying $|2i-2n-1|+|2j-2n-1| \leq 2n$ and $m_{i,j} = 0$ for all other $i,j$, consisting of a central diamond of 1's surrounded by 0's. When $n \geq…

Let $A$ be a finite subset of a field $\mathbb{F}$ and $D_n(A)$ be a set of all matrices with entries in $A$, namely $$ D_n(A)=\{D\in \mathbb{F}\ |\ \exists a_{ij}\in A, 1 \le i,j \le n, \det\bigl((a_{ij})\bigr)=D\}, $$ where the symbol…

A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite…

An $n\times n$ matrix $M$ is called a \textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\ell} M_{\ell,k} = 0$ for every $k\ne \ell$. Dietzfelbinger, Hromkovi{\v{c}}, and Schnitger (1996) showed that $n…

In this short note we prove a lemma about the dimension of certain algebraic sets of matrices. This result is needed in our paper arXiv:1201.1672. The result presented here has also applications in other situations and so it should appear…

We investigate the existence of heavy columns in binary matrices with distinct rows. A column of an m x n binary matrix is called heavy if the number of ones in it is at least m/2. We introduce two recursive algorithms, A1 and A2, that…

Let ${\bf A}={\bf A}_{n,m,k}$ be a random $n\times m$ matrix over $\mathbf{GF}_2$ wher each column consists of $k$ randomly chosen ones. Let $M$ be an arbirary fixed binary matroid. We show that if $m/n$ and $k$ are sufficiently large then…

In the work are defined the concepts semi-canonical and canonical binary matrix. What is described is an algorithm solving the combinatorial problem for finding the semi-canonical matrices in the set \Lambda_n^k consisting of all n\times n…

Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank $r$ and with bounded subdeterminants. In particular, we study the…

Given $n$ items with at most $d$ of which being positive, instead of testing these items individually, the theory of combinatorial group testing aims to identify all positive items using as few tests as possible. This paper is devoted to a…

We prove, by means of an exact structural description, that every simple triangle-free binary matroid $M$ with $|M| > \tfrac{33}{128}2^{r(M)}$ has critical number at most $2$.

A theorem of Mina evaluates the determinant of a matrix with entries $D^j(f(x)^i)$. We note the important special case where the matrix entries are evaluated at $x=0$ and give a simple proof of it, and some applications. We then give a…

We study the algebraic matroid induced by the ideal of (r+1)-minors of a matrix of variables over a field. This is inherently connected to the bounded-rank matrix completion problem, in which the aim is to complete a partially observed rank…