Technology Category
Reasoning under Uncertainty: Stochastic OptimizationConstraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationKnowledge Representation and Reasoning: Computational Complexity of Reasoning
A parameterized linear formulation of the integer hull
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Two-stage Stochastic Integer Programming
Polyhedral Description
Integer Envelope
Innovation
Methods, ideas, or system contributions that make the work stand out.
Integer Envelope
Gomory-Chvatal Cut Plane Theory
Rational Polyhedral Construction
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