Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics

We systematically investigate the connections between constraint satisfaction problems, structures of bounded treewidth, and definability in logics with a finite number of variables. We first show that constraint satisfaction problems on inputs of treewidth less than k are definable using Datalog programs with at most k variables; this provides a new explanation for the tractability of these classes of problems. After this, we investigate constraint satisfaction on inputs that are homomorphically equivalent to structures of bounded treewidth. We show that these problems are solvable in polynomial time by establishing that they are actually definable in Datalog; moreover, we obtain a logical characterization of the property “being homomorphically equivalent to a structure of bounded treewidth” in terms of definability in finite-variable logics. Unfortunately, this expansion of the tractability landscape comes at a price, because we also show that, for each k ⩾ 2, determining whether a structure is homomorphically equivalent to a structure of treewidth less than k is an NP-complete problem. In contrast, it is well known that, for each k ⩾ 2, there is a polynomial-time algorithm for testing whether a given structure is of treewidth less than k. Finally, we obtain a logical characterization of the property “having bounded treewidth” that sheds light on the complexity-theoretic difference between this property and the property ‘being homomorphically equivalent to a structure of bounded treewidth”.