Parametric integer programming in fixed dimension

We consider the following problem: Given a rational matrix $A \in \mathbb{Q}^{m \times n}$ and a rational polyhedron $Q \subseteq \mathbb{R}^{m+p}$, decide if for all vectors $b \in \mathbb{R}^m$, for which there exists an integral $z \in \mathbb{Z}^p$ 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 $\max \{ c^T x : A x \leq b, x \in \mathbb{Z}^n \}$ 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}sten and Sturmfels (2003) who presented such an algorithm for integer programs in standard form.