14 problems

Let $A\in\mathbb{R}_{+}^{m\times n}$ be a nonnegative matrix, where $\operatorname{rk}(A)$ denotes its ordinary rank and $\operatorname{rk}_{+}(A)$ its nonnegative rank. Vandaele–G…

Let $A_1$ and $A_2$ be nonnegative matrices. Dagstuhl multiplicativity conjecture. It was conjectured that … This conjecture was later disproved by constructing nonnegative matrice…

For $n\in\mathbb{N}$, let $C_n\in\mathbb{R}_+^{2^n\times 2^n}$ be defined by … for $a,b\in\{0,1\}^n$, and let $\operatorname{rank}_+(C_n)$ denote its nonnegative rank. Full nonnega…

Let $M$ be a $0/1$-matrix, and let $\mathrm{rk}(M)$ denote its rank. A deterministic communication protocol computes $M$ when it determines every matrix entry, and the nonnegative…

A rational Diophantine equation is a polynomial equation with rational coefficients whose variables are required to take rational values. Undecidability conjecture. There is no alg…

Generic maximal-rank conjecture. The combinatorial hypersimplices of nonnegative rank $2n$ form a dense open subset of $\mathcal{R}_{n,k}$.

Let $P_1$ and $P_2$ be polytopes. Their nonnegative rank is the smallest number of nonnegative factors in a nonnegative factorization of a slack matrix of the polytope. Product add…

Local stabilization conjecture. For given $r\geq 3$ there exists an $n_0 \in \mathbb{N}$, such that for all $n\geq n_0$ and for all nonnegative factorizations $M=AB$ where $M$ is o…

Algebraic-boundary stabilization conjecture. For given $r \geq 3$ there exist $n_0 \in \mathbb{N}$, such that for all $n\geq n_0$ and for all matrices $M=(m_i)_{i=1,...,n}$ on the…

Correlation-polytope submatrix conjecture. The matrix $M_n$ has full nonnegative rank:

Let $S_n$ be the slack matrix of any $n$-gon, and let $\lfloor x\rfloor$ denote the largest integer not exceeding $x$. Polygon slack-matrix upper-bound conjecture. … and equality h…

Let $A$ be a nonnegative matrix, let $A\otimes A$ denote its Kronecker square, and write $\operatorname{rank}(A)$ for its ordinary rank. Self-Kronecker-product conjecture. … The so…

For a nonnegative matrix $M$, its nonnegative rank $\operatorname{rank}_+(M)$ is the least inner dimension of a factorization of $M$ into two nonnegative matrices. The ordinary ran…

Let $M$ be a random linear Euclidean distance matrix of dimension $n$, so that $M$ is an $n\times n$ matrix of pairwise squared distances generated by randomly chosen points on a l…