A parameterized linear formulation of the integer hull

Abstract

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.