The power of tree projections: local consistency, greedy algorithms, and larger islands of tractability

Enforcing local consistency is a well-known technique to simplify the evaluation of conjunctive queries. It consists of repeatedly taking the semijion between every pair of (relations associated with) query atoms, until the procedure stabilizes. If some relation becomes empty, then the query has an empty answer. Otherwise, we cannot say anything in general, unless we have some information on the structure of the given query. In fact, a fundamental result in database theory states that the class of queries for which---on every database---local consistency entails global consistency is precisely the class of acyclic queries. In the last few years, several efforts have been made to define structural decomposition methods isolating larger classes of nearly-acyclic queries, yet retaining the same nice properties as acyclic ones. In particular, it is known that queries having bounded (generalized) hypertree-width can be evaluated in polynomial time, and that this structural property is also sufficient to guarantee that local consistency solves the problem, as for acyclic queries. However, the precise power of such an approach was an open problem: Is it the case that bounded generalized hypertree-width is also a necessary condition to guarantee that local consistency entails global consistency? In this paper, we positively answer this question, and go beyond. Firstly, we precisely characterize the power of local consistency procedures in the more general framework of tree projections, where a query Q and a set V of views (i.e., resources that can be used to answer Q) are given, and where one looks for an acyclic hypergraph covering Q and covered by Q---all known structural decomposition methods are just special cases of this framework, defining their specific set of resources. We show that the existence of tree projections of certain subqueries is a necessary and sufficient condition to guarantee that local consistency entails global consistency. In particular, tight characterizations are given not only for the decision problem, but also when answers restricted to variables covered by some view have to be computed. Secondly, we consider greedy tree-projections that are easy to compute, and we study how far they can be from arbitrary tree-projections, which are intractable in general. Finally, we investigate classes of instances not included in those having tree projections, and which can be easily recognized and define either new islands of tractability, or islands of quasi-tractability.