Deep Weisfeiler Leman
share
We introduce the framework of Deep Weisfeiler Leman algorithms (DeepWL), which allows the design of purely combinatorial graph isomorphism tests that are more powerful than the well-known Weisfeiler-Leman algorithm. We prove that, as an abstract computational model, polynomial time DeepWL-algorithms have exactly the same expressiveness as the logic Choiceless Polynomial Time (with counting) introduced by Blass, Gurevich, and Shelah (Ann. Pure Appl. Logic., 1999) It is a well-known open question whether the existence of a polynomial time graph isomorphism test implies the existence of a polynomial time canonisation algorithm. Our main technical result states that for each class of graphs (satisfying some mild closure condition), if there is a polynomial time DeepWL isomorphism test then there is a polynomial canonisation algorithm for this class. This implies that there is also a logic capturing polynomial time on this class.
READ FULL TEXT