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We consider the following problem: Given a rational matrix A ∈ Qm×n and a rational polyhe- dron 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 É 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 who established such an algorithm for the case when, in addition to p and n, the affine dimension ofQ 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 É b, x ∈ Zn ª and its lin- ear programming relaxation over all right-hand sides b, for which the integer program is fea- sible. The algorithm is polynomial if n is fixed. This is an extension of a recent result of Hosten and Sturmfels who presented such analgorithmforinteger programs instandardform.

Parametric Integer Programming in Fixed Dimension