Parametric Integer Programming in Fixed Dimension

We consider the following problem: Given a rational matrix A ∈ Qm×n and a rational polyhedron Q ⊆ Rm+p, decide if for all vectors b ∈ Rm, for which there exists an integral z ∈ Zp such that (b,z) ∈ Q, the system of linear inequalities Ax 6 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{cx : Ax 6 b, x ∈ Zn} 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 Hoşten and Sturmfels (2003) who presented such an algorithm for integer programs in standard form.