Bounded width problems and algebras

Abstract.Let $${\mathcal A}$$ be finite relational structure of finite type, and let CSP $${(\mathcal A)}$$ denote the following decision problem: if $${\mathcal I}$$ is a given structure of the same type as $${\mathcal A}$$ , is there a homomorphism from $${\mathcal I}$$ to $${\mathcal A}$$ ? To each relational structure $${\mathcal A}$$ is associated naturally an algebra $${\mathbb A}$$ whose structure determines the complexity of the associated decision problem. We investigate those finite algebras arising from CSP’s of so-called bounded width, i.e., for which local consistency algorithms effectively decide the problem. We show that if a CSP has bounded width then the variety generated by the associated algebra omits the Hobby-McKenzie types 1 and 2. This provides a method to prove that certain CSP’s do not have bounded width. We give several applications, answering a question of Nešetřil and Zhu [26], by showing that various graph homomorphism problems do not have bounded width. Feder and Vardi [17] have shown that every CSP is polynomial-time equivalent to the retraction problem for a poset we call the Feder − Vardi poset of the structure. We show that, in the case where the structure has a single relation, if the retraction problem for the Feder-Vardi poset has bounded width then the CSP for the structure also has bounded width. This is used to exhibit a finite order-primal algebra whose variety admits type 2 but omits type 1 (provided P ≠ NP).