Variation of cost functions in integer programming

We study the problem of minimizing c x subject to A x = b, x 0 and x integral, for a xed matrix A. Two cost functions c and c 0 are considered equivalent if they give the same optimal solutions for each b. We construct a polytope St(A) whose normal cones are the equivalence classes. Explicit inequality presentations of these cones are given by the reduced Grr obner bases associated with A. The union of the reduced Grr obner bases as c varies (called the universal Grr obner basis) consists precisely of the edge directions of St(A). We present geometric algorithms for computing St(A), the Graver basis Gra], and the universal Grr obner basis.