A copositive formulation for the stability number of infinite graphs

TL;DR

This paper extends copositive formulations to infinite graphs, establishing a duality theory and providing a new approach to compute the stability number using infinite-dimensional optimization techniques.

Contribution

It introduces a novel copositive formulation for the stability number of infinite graphs and develops a duality theory involving copositive kernels and completely positive measures.

Findings

Established a duality between copositive kernels and completely positive measures.

Determined the extreme rays of the cone of completely positive measures.

Applied the theory to the kissing number problem.

Abstract

In the last decade, copositive formulations have been proposed for a variety of combinatorial optimization problems, for example the stability number (independence number). In this paper, we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite-dimensional copositive program. For this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures. We determine the extreme rays of the latter cone, and we illustrate this theory with the help of the kissing number problem.