A parameterized linear formulation of the integer hull

TL;DR

This paper introduces a parameterized linear formulation for the integer hull of polyhedra defined by integer matrices, enabling affine descriptions within certain lattice classes and solving an open problem in stochastic integer programming.

Contribution

It proves that the integer hull can be described by an affine function of the right-hand side within fixed lattice classes, with computable parameters depending only on matrix bounds and dimension.

Findings

Affine description of integer hull within lattice classes.

Polynomial-time computability of key parameters.

Application to complexity of 2-stage stochastic integer programming.

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.