Finite-level semidefinite characterization of commutativity gadgets

Determine whether there exists a hierarchy of semidefinite relaxations for projective packings in which the existence of commutativity gadgets for graphs is characterized by a suitable rigidity condition at some finite level.

Background

The paper develops a spectral method for proving that all quantum polymorphisms of certain graphs are non-contextual, thereby establishing the existence of commutativity gadgets and the RE-completeness of the associated quantum graph homomorphism problems. The method relies on Schrijver-rigidity, which is an equality condition for Schrijver's semidefinite bound on projective packings, together with structural properties of disjointness representations.

The authors explain that their rigidity criterion does not capture every graph admitting a commutativity gadget: odd cycles of length at least five provide examples that possess commutativity gadgets but do not satisfy the paper's Schrijver-rigidity condition. They therefore ask whether a broader spectral framework, formulated as a hierarchy of semidefinite relaxations, could characterize commutativity-gadget existence at some finite level.