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A parameterized linear formulation of the integer hull

Friedrich Eisenbrand, Thomas Rothvoss

TL;DR

The paper addresses the problem of describing the integer hull of parameterized systems $P(b)=\{x: Ax\le b\}$ with a bounded coefficient matrix $A$ by showing that, within each lattice class modulo a computed $D$, the right-hand side of the linear description can be made affine in $b$. The authors develop a parametric, modulus-based extension of Chvátal-Gomory cutting planes via the elementary closure, proving that there exist $D$, $B$, $C$, and residue-dependent vectors $f_r$ such that $P(b)_I = \{ x: Bx \le f_r + C b \}$ whenever $b-r \in D\mathbb{Z}^m$. They provide explicit, triple-exponential bounds on $D$ and show that both $D$ and the associated matrices can be computed in time depending only on $n$ and $\Delta$, with the non-repeating-rows assumption removable. The framework yields practical consequences for open questions in 2-stage stochastic IPs, the convexity of integer cones, and proximity-based approaches to the 4-block problem, enabling fixed-parameter tractable algorithms and almost-feasible solutions in structured IP families.

Abstract

Let $A \in \mathbb{Z}^{m \times n}$ be an integer matrix with components bounded by $Δ$ 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 $Δ$ only. Furthermore, $D$ as well as the matrix $B$ can be computed in time depending on $Δ$ 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{Chvátal-Gomory cutting planes} and the \emph{elementary closure} of rational polyhedra.

A parameterized linear formulation of the integer hull

TL;DR

The paper addresses the problem of describing the integer hull of parameterized systems with a bounded coefficient matrix by showing that, within each lattice class modulo a computed , the right-hand side of the linear description can be made affine in . The authors develop a parametric, modulus-based extension of Chvátal-Gomory cutting planes via the elementary closure, proving that there exist , , , and residue-dependent vectors such that whenever . They provide explicit, triple-exponential bounds on and show that both and the associated matrices can be computed in time depending only on and , with the non-repeating-rows assumption removable. The framework yields practical consequences for open questions in 2-stage stochastic IPs, the convexity of integer cones, and proximity-based approaches to the 4-block problem, enabling fixed-parameter tractable algorithms and almost-feasible solutions in structured IP families.

Abstract

Let be an integer matrix with components bounded by in absolute value. Cook et al.~(1986) have shown that there exists a universal matrix with the following property: For each , there exists such that the integer hull of the polyhedron is described by . Our \emph{main result} is that is an \emph{affine} function of as long as is from a fixed equivalence class of the lattice . Here is a number that depends on and only. Furthermore, as well as the matrix can be computed in time depending on and 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{Chvátal-Gomory cutting planes} and the \emph{elementary closure} of rational polyhedra.
Paper Structure (11 sections, 12 theorems, 48 equations, 1 figure)

This paper contains 11 sections, 12 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Chvátal-Gomory cutting planes
  3. Proof of the main theorem
  4. Warmup --- Linearity of the first elementary closure
  5. Linearity of the $i$-th elementary closure
  6. Applications
  7. The convexity of integer cones
  8. Optimizing over 2-stage stochastic IPs
  9. An almost solution to the 4-block problem
  10. Acknowledgements.
  11. A proximity bound for the integer hull

Key Result

Theorem 1

Let $A ∈ℤ^{m ×n}$ with non-repeating rows and $\|A\|_∞ ≤Δ$. There exists a $D ∈ ℕ$, and matrices $B ∈ℤ^{m'×n}$ and $C ∈ℤ^{m'×m}$ depending on $A$ such that, for each $r ∈ \{0,\dots,D-1\}^m$, there exist an $f_r∈ℤ^{m'}$ such that the following holds: For each $b \in \mathbb{Z}^m$ with $b - r ∈ {D ⋅ ℤ

Figures (1)

  • Figure 1: The valid inequality $-x_1+x_2 ≤ δ$ yields the cutting plane $-x_1+x_2 ≤ ⌊δ ⌋$.

Theorems & Definitions (23)

  • Theorem 1
  • Remark 1
  • Theorem 2: Schrijver Schrijver80
  • proof
  • Corollary 3
  • Theorem 4: Cook, Gerards, Schrijver and Tardos MR839604
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • ...and 13 more