A parameterized linear formulation of the integer hull

2025-01-04

AI Summary

This paper addresses the open complexity question posed by Cslovjecsek et al. (2024) on two-stage stochastic integer programming, focusing on the parametric linear characterization of the integer hull when the right-hand side vector (b) varies. Method: Leveraging Gomory–Chvátal cutting planes, rational polyhedral elementary closures, lattice theory, and parametric integer programming, we analyze the integer hull under fixed lattice-equivalence classes of (b). Contribution/Results: We establish that, within each such class, the integer hull admits an exact linear description via a system of inequalities affine in (b). Crucially, the constraint matrix (B) depends only on the variable dimension (n) and coefficient magnitude bound (Delta), while the right-hand side vector (t) is an affine function of (b). This yields the first unified, computable, and fixed-parameter tractable (FPT)-constructible parametric linear representation of the integer hull. Both (B) and the parameter (D) can be computed in time polynomial solely in (n) and (Delta), providing the first FPT algorithmic framework for two-stage stochastic integer programming.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Large-scale security measurements

Abstract

Let $A in mathbb{Z}^{m imes n}$ be an integer matrix with components bounded by $Delta$ in absolute value. Cook et al.~(1986) have shown that there exists a universal matrix $B in mathbb{Z}^{m' imes n}$ 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{Gomory-Chv'atal cutting planes} and the emph{elementary closure} of rational polyhedra.

Problem

• Two-stage Stochastic Integer Programming
• Polyhedral Description
• Integer Envelope

Innovation

• Integer Envelope
• Gomory-Chvatal Cut Plane Theory
• Rational Polyhedral Construction