Copositive matrices, sums of squares and the stability number of a graph

TL;DR

This paper explores copositive matrices and their sums of squares representations to improve bounds on the maximum stable set problem in graphs, analyzing when these approximations are exact.

Contribution

It develops conic inner approximations of the copositive cone using sums of squares, enhancing understanding of their accuracy for graph stability number calculations.

Findings

Conic approximations can exactly represent the copositive cone in certain cases.

Sum of squares certificates provide effective positivity conditions for copositivity.

Finite convergence of bounds depends on specific graph and matrix properties.

Abstract

This chapter investigates the cone of copositive matrices, with a focus on the design and analysis of conic inner approximations for it. These approximations are based on various sufficient conditions for matrix copositivity, relying on positivity certificates in terms of sums of squares of polynomials. Their application to the discrete optimization problem asking for a maximum stable set in a graph is also discussed. A central theme in this chapter is understanding when the conic approximations suffice for describing the full copositive cone, and when the corresponding bounds for the stable set problem admit finite convergence.