Inverse of the Gomory Corner Relaxation of Integer Programs

We explore the inverse of integer programs (IPs) by studying the inverse of their Gomory corner relaxations (GCRs). We show that solving a set of inverse GCR problems always yields an upper bound on the optimal value of the inverse IP that is at least as tight as the optimal value of the inverse of the linear program (LP) relaxation. We provide conditions under which solving a set of inverse GCR problems exactly solves the inverse IP. We propose an LP formulation for solving the inverse GCR under the $L_1$ and $L_\infty$ norms by reformulating the inverse GCR as the inverse of a shortest path problem.