3 problems

Let $A$ be an integer matrix with columns ${\boldsymbol v}_1,\ldots,{\boldsymbol v}_n\in\mathbb{Z}^m$, let … and set ${\boldsymbol v}={\boldsymbol v}_1+\cdots+{\boldsymbol v}_n$. L…

For each integer $m\geq 2$, consider the integer program $(\mathrm{CE})_m$ with variables $x\in\{0,1\}^{2m}$ and its linear relaxation. Let $x_m^\circ$ denote an optimal solution o…

Consider an instance of the multiple knapsack problem with compacity constraints, its linear relaxation $(\mathrm{mKPC})_{\mathrm{LP}}$, and its naive semidefinite relaxation. Writ…