Does query evaluation tractability help query containment?

Abstract

While checking containment of Datalog programs is undecidable, checking whether a Datalog program is contained in a union of conjunctive queries (UCQ), in the context of relational databases, or a union of conjunctive 2-way regular path queries (UC2RPQ), in the context of graph databases, is decidable. The complexity of these problems is, however, prohibitive: 2exptime-complete. We investigate to which extent restrictions on UCQs and UC2RPQs, which have been known to reduce the complexity of query containment for these classes, yield a more "manageable" single-exponential time bound, which is the norm for several static analysis and verification tasks. Checking containment of a UCQ Theta' in a UCQ Theta is NP-hard, in general, but better bounds can be obtained if Theta is restricted to belong to a "tractable" class of UCQs, e.g., a class of bounded treewidth or hypertreewidth. Also, each Datalog program Pi is equivalent to an infinite union of CQs. This motivated us to study the question of whether restricting Theta to belong to a tractable class also helps alleviate the complexity of checking whether Pi is contained in Theta. We study such question in detail and show that the situation is much more delicate than expected: First, tractability of UCQs does not help in general, but further restricting Theta to be acyclic and have a bounded number of shared variables between atoms yields better complexity bounds. As corollaries, we obtain that checking containment of Pi in Theta is in exptime if Theta is of treewidth one, or it is acyclic and the arity of the schema is fixed. In the case of UC2RPQs we show an exptime bound when queries are acyclic and have a bounded number of edges connecting pairs of variables. As a corollary, we obtain that checking whether Pi is contained in UC2RPQ Gamma is in exptime if Gamma is a strongly acyclic UC2RPQ. Our positive results for UCQs and UC2RPQs are optimal, in a sense, since slightly extending the conditions turns the problem 2exptime-complete.