Integer matrices that are not copositive have certificates of less than quadratic complexity

TL;DR

This paper establishes that non-copositive integer matrices possess certificates of negativity with complexity bounded by a polynomial function of their binary size, improving understanding of their certification properties.

Contribution

The paper proves that certificates for non-copositive integer matrices have complexity less than quadratic in the binary encoding length, providing a new bound on certificate complexity.

Findings

Certificates have complexity less than quadratic in the binary size of the matrix.

Guarantees existence of certificates within polynomial bounds for integer matrices.

Enhances understanding of the certification process for non-copositive matrices.

Abstract

A real symmetric n times n matrix is called copositive if the corresponding quadratic form is non-negative on the closed first orthant. If the matrix fails to be copositive there exists some non-negative certificate for which the quadratic form is negative. Due to the scaling property, we can find such certificates in every neighborhood of the origin but their properties depend on the matrix of course and are hard to describe. If it is an integer matrix however, we are guaranteed certificates of a complexity that is at most a constant times the binary encoding length of the matrix raised to the power 3/2.