Fractional homomorphism, Weisfeiler-Leman invariance, and the Sherali-Adams hierarchy for the Constraint Satisfaction Problem

Abstract

Given a pair of graphs $\textbf{A}$ and $\textbf{B}$, the problems of deciding whether there exists either a homomorphism or an isomorphism from $\textbf{A}$ to $\textbf{B}$ have received a lot of attention. While graph homomorphism is known to be NP-complete, the complexity of the graph isomorphism problem is not fully understood. A well-known combinatorial heuristic for graph isomorphism is the Weisfeiler-Leman test together with its higher order variants. On the other hand, both problems can be reformulated as integer programs and various LP methods can be applied to obtain high-quality relaxations that can still be solved efficiently. We study so-called fractional relaxations of these programs in the more general context where $\textbf{A}$ and $\textbf{B}$ are not graphs but arbitrary relational structures. We give a combinatorial characterization of the Sherali-Adams hierarchy applied to the homomorphism problem in terms of fractional isomorphism. Collaterally, we also extend a number of known results from graph theory to give a characterization of the notion of fractional isomorphism for relational structures in terms of the Weisfeiler-Leman test, equitable partitions, and counting homomorphisms from trees. As a result, we obtain a description of the families of CSPs that are closed under Weisfeiler-Leman invariance in terms of their polymorphisms as well as decidability by the first level of the Sherali-Adams hierarchy.