Bounded width problems and algebras

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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 Nesetřil 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).