An Algebraic Approach to the Complexity of Generalized Conjunctive Queries

Conjunctive-query containment is considered as a fundamen- tal problem in database query evaluation and optimization. Kolaitis and Vardi pointed out that constraint satisfaction and conjunctive query con- tainment are essentially the same problem. We study the Boolean con- junctive queries under a more detailed scope, where we investigate their counting problem by means of the algebraic approach through Galois theory, taking advantage of Post's lattice. We prove a trichotomy the- orem for the generalized conjunctive query counting problem, showing this way that, contrary to the corresponding decision problems, con- straint satisfaction and conjunctive-query containment differ for other computational goals. We also study the audit problem for conjunctive queries asking whether there exists a frozen variable in a given query. This problem is important in databases supporting statistical queries. We derive a dichotomy theorem for this audit problem that sheds more light on audit applicability within database systems.