Parametric Integer Programming in Fixed Dimension

TL;DR

This paper presents a polynomial-time algorithm for parametric integer programming in fixed dimensions, extending previous results and providing new methods for analyzing the difference between integer and linear program optima.

Contribution

It extends Kannan's 1990 algorithm to fixed p and n, and offers a new polynomial-time approach for parametric integer programming with applications to integer program bounds.

Findings

Algorithm solves the problem in polynomial time for fixed p and n.

Extends Kannan's 1990 result to broader fixed-dimension cases.

Provides an algorithm to compute maximum differences between integer and LP optima.

Abstract

We consider the following problem: Given a rational matrix $A \in \setQ^{m \times n}$ and a rational polyhedron $Q \subseteq\setR^{m+p}$, decide if for all vectors $b \in \setR^m$, for which there exists an integral $z \in \setZ^p$ such that $(b, z) \in Q$, the system of linear inequalities $A x \leq b$ has an integral solution. We show that there exists an algorithm that solves this problem in polynomial time if $p$ and $n$ are fixed. This extends a result of Kannan (1990) who established such an algorithm for the case when, in addition to $p$ and $n$, the affine dimension of $Q$ is fixed. As an application of this result, we describe an algorithm to find the maximum difference between the optimum values of an integer program $\max \{c x : A x \leq b, x \in \setZ^n \}$ and its linear programming relaxation over all right-hand sides $b$, for which the integer program is feasible. The algorithm is polynomial if $n$ is fixed. This is an extension of a recent result of Ho\c{s}ten and Sturmfels (2003) who presented such an algorithm for integer programs in standard form.