Certificates and relaxations for integer programming and the semi-group membership problem

TL;DR

This paper introduces a theorem of the alternative linking integer programming and semi-group membership, along with a hierarchy of LP relaxations to analyze solutions.

Contribution

It presents a novel theorem of the alternative for integer solutions and characterizes convex cones for polynomial coefficients, advancing the understanding of these problems.

Findings

The system Ax=b has no nonnegative integer solution iff p(b)<0 for some polynomial p.

A hierarchy of LP relaxations is developed for the problem.

The first relaxation in the hierarchy corresponds to the continuous case Ax=b with real x.

Abstract

We consider integer programming and the semi-group membership problem. We provide the following theorem of the alternative: the system Ax=b has no nonnegative integral solution x if and only if p(b) <0 for some given polynomial p whose vector of coefficients lies in a convex cone that we characterize. We also provide a hierarchy of linear programming relaxations, where the continuous case Ax=b with x real and nonnegative, describes the first relaxation in the hierarchy.