Bounded width problems and algebras

Let be finite relational structure of finite type, and let CSP denote the following decision problem: if is a given structure of the same type as , is there a homomorphism from to ? To each relational structure is associated naturally an algebra 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). Presented by M. Valeriote. Received January 8, 2005; accepted in final form April 3, 2006. The first author’s research is supported by a grant from NSERC and the Centre de Recherches Mathématiques. The second author’s research is supported by OTKA no. 034175 and 48809 and T 037877. Part of this research was conducted while the second author was visiting Concordia University in Montréal and also when the first author was visiting the Bolyai Institute in Szeged. The support of NSERC, OTKA and the Bolyai Institute is gratefully acknowledged.