Let $A \in \mathbb{Z}^{m \times n}$ be an integer matrix with components bounded by $\Delta$ in absolute value. Cook et al.~(1986) have shown that there exists a universal matrix $B \in \mathbb{Z}^{m' \times n}$ with the following property: For each $b \in \mathbb{Z}^m$, there exists $t \in \mathbb{Z}^{m'}$ such that the integer hull of the polyhedron $P = \{ x \in \mathbb{R}^n \colon Ax \leq b\}$ is described by $P_I = \{ x \in \mathbb{R}^n \colon Bx \leq t\}$. Our \emph{main result} is that $t$ is an \emph{affine} function of $b$ as long as $b$ is from a fixed equivalence class of the lattice $D \cdot \mathbb{Z}^m$. Here $D \in \mathbb{N}$ is a number that depends on $n$ and $\Delta$ only. Furthermore, $D$ as well as the matrix $B$ can be computed in time depending on $\Delta$ and $n$ only. An application of this result is the solution of an open problem posed by Cslovjecsek et al.~(SODA 2024) concerning the complexity of \emph{2-stage-stochastic integer programming} problems. The main tool of our proof is the classical theory of \emph{Chv\'atal-Gomory cutting planes} and the \emph{elementary closure} of rational polyhedra.

翻译：设 $A \in \mathbb{Z}^{m \times n}$ 为整数矩阵，其各分量的绝对值以 $\Delta$ 为界。Cook 等人（1986）已证明存在一个通用矩阵 $B \in \mathbb{Z}^{m' \times n}$ 满足以下性质：对任意 $b \in \mathbb{Z}^m$，均存在 $t \in \mathbb{Z}^{m'}$，使得多面体 $P = \{ x \in \mathbb{R}^n \colon Ax \leq b\}$ 的整数包络可表示为 $P_I = \{ x \in \mathbb{R}^n \colon Bx \leq t\}$。我们的主要结果表明：只要 $b$ 取自格 $D \cdot \mathbb{Z}^m$ 的固定等价类，$t$ 即为 $b$ 的仿射函数。此处 $D \in \mathbb{N}$ 是仅依赖于 $n$ 和 $\Delta$ 的常数。进一步地，$D$ 与矩阵 $B$ 的计算时间复杂度仅取决于 $\Delta$ 和 $n$。该结果的一个应用是解决了 Cslovjecsek 等人（SODA 2024）提出的关于两阶段随机整数规划问题复杂度的公开问题。证明的核心工具是经典的 Chvátal-Gomory 割平面理论与有理多面体的基本闭包理论。