Inverse of the Gomory Corner Relaxation of Integer Programs

TL;DR

This paper investigates the inverse of Gomory corner relaxations for integer programs, providing bounds, conditions for exact solutions, and LP formulations under specific norms, advancing inverse optimization methods.

Contribution

It introduces a novel approach to inverse Gomory corner relaxations, including bounds, exact solution conditions, and LP reformulations for inverse problems.

Findings

Inverse GCR solutions provide bounds at least as tight as inverse LP relaxations.

Conditions are identified under which inverse GCR solutions exactly solve the inverse IP.

LP formulations for inverse GCR under $L_1$ and $L_inity$ norms are developed.

Abstract

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.