10 problems

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…

For each positive integer $n$, let $\mathsf{S}_n$ be the sparse paving matroid on $[n]$ of rank $\lfloor n/2\rfloor$ with the maximal possible number of circuit-hyperplanes and lex…

Let $\mathcal{G}$ be a proper minor-closed graph class, and let $G$ be a connected $n$-vertex graph in $\mathcal{G}$. The spanning tree polytope of $G$ is the convex hull of the in…

Let $\mathcal{C}$ be a proper minor-closed family of graphs, and let $G=(V,E)$ be a connected graph in $\mathcal{C}$. Let $\operatorname{P_{\mathrm{sp.trees}}}(G)$ denote the conve…

Let $G$ be a graph in the explicit family constructed in the paper, let $n$ be its number of nodes, and let $P_G$ denote the convex hull of the indicator vectors of the independent…

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

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…

A polytope $P$ has positive semidefinite extension complexity equal to the smallest $r$ such that $P$ is the projection of an affine slice of the cone of $r$-dimensional positive s…

Let the perfect matching polytope be the polytope whose vertices correspond to perfect matchings of a graph, and let a linear extension formulation (EF) be an extended linear descr…