10 problems

Sturmfels–Sullivant's conjecture. The following statements are equivalent:

Collins–Gisin conjecture. For every $m\geq 1$, the $\mathrm{I}_{mm22}$ inequality is a facet of $\mathrm{COR}^\square(\mathrm{K}_{m,m})$; equivalently, it is a tight Bell inequalit…

Exactness conjecture. The second semidefinite lifting is exact for $\mathrm{CUT}^4_\infty$, i.e.,

Let $G(V,E)$ be a graph, and let $\operatorname{Cut}^\square(G)$ be the convex hull of the cut points $\delta_{A\vert B}\in\mathbb{R}^{\lvert E\rvert}$ over all unordered vertex pa…

Facet-orbit conjecture. If $G=M_{2m}$ with $m\ge 4$, then among the facets of $\operatorname{CUTP}(G)$ there are two orbits of $4m$ and $2m$ edge facets from the ring- and rung-edg…

Let $\operatorname{CUTP}_n$ denote the cut polytope on the complete graph with $n$ vertices, and call two facets adjacent when they share a ridge. A triangle inequality facet is a…

Let $n$ be odd, and consider the cut polytope of the complete graph $K_n$. The Lasserre rank is the least level at which the Lasserre semidefinite relaxation is exact. Laurent's co…

Normality–$K_5$-minor conjecture. The following conditions are equivalent:

The grid-graph conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if $G$ is a grid graph.

The 4-connected plane triangulation conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if $G$ is a 4-connected plane triangulation.