Parametric Integer Programming in Fixed Dimension
arXiv:0801.4336
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 , the system of linear inequalities has an integral solution. We show that there exists an algorithm that solves this problem in polynomial time if and are fixed. This extends a result of Kannan (1990) who established such an algorithm for the case when, in addition to and , the affine dimension of 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 , for which the integer program is feasible. The algorithm is polynomial if is fixed. This is an extension of a recent result of Hoşten and Sturmfels (2003) who presented such an algorithm for integer programs in standard form.
23 pages, 3 figures