The Quadratic Graver Cone, Quadratic Integer Minimization, and Extensions

TL;DR

This paper introduces a polynomial-time method for minimizing certain quadratic functions over integer points, leveraging the Graver basis and dual Graver cone, with extensions to polynomial functions of arbitrary degree.

Contribution

It establishes a polynomial-time algorithm for quadratic integer minimization using the dual Graver cone and explores its relation to positive semidefinite matrices, extending to higher-degree polynomials.

Findings

Polynomial-time solvability for some non-convex quadrics

Relation between dual Graver cone and positive semidefinite cone

Extension to arbitrary degree polynomial minimization

Abstract

We consider the nonlinear integer programming problem of minimizing a quadratic function over the integer points in variable dimension satisfying a system of linear inequalities. We show that when the Graver basis of the matrix defining the system is given, and the quadratic function lies in a suitable {\em dual Graver cone}, the problem can be solved in polynomial time. We discuss the relation between this cone and the cone of positive semidefinite matrices, and show that none contains the other. So we can minimize in polynomial time some non-convex and some (including all separable) convex quadrics. We conclude by extending our results to efficient integer minimization of multivariate polynomial functions of arbitrary degree lying in suitable cones.