Computing the integer points of a polyhedron

The integer points of polyhedral sets are of interest in many areas of mathematical sciences, see for instance the landmark textbooks of A. Schrijver [18] and A. Barvinok [3], as well as the compilation of articles [4]. One of these areas is the analysis and transformation of computer programs. For instance, integer programming [6] is used by P. Feautrier in the scheduling of for-loop nests [7], Barvinok's algorithm [2] for counting integer points in polyhedra is adapted by M. Köppe and S. Verdoolaege in [15] to answer questions like how many memory locations are touched by a for-loop nest. In [16], W. Pugh proposes an algorithm, called the Omega Test, for testing whether a polyhedron has integer points. In the same paper, W. Pugh shows how to use the Omega Test for performing dependence analysis [16] in for-loop nests. In [17], W. Pugh also suggests, without stating a formal algorithm, that the Omega Test could be used for quantifier elimination on Presburger formulas. This observation is a first motivation for the work presented here.