On Acyclic Conjunctive Queries and Constant Delay Enumeration

Abstract

We study the enumeration complexity of the natural extension of acyclic conjunctive queries with disequalities. In this language, a number of NP-complete problems can be expressed. We first improve a previous result of Papadimitriou and Yannakakis by proving that such queries can be computed in time $c.|\cal M|.|\varphi(\cal M)|$ where $\cal M$ is the structure, $\varphi(\cal M)$ is the result set of the query and c is a simple exponential in the size of the formula ϕ. A consequence of our method is that, in the general case, tuples of such queries can be enumerated with a linear delay between two tuples.

We then introduce a large subclass of acyclic formulas called CCQ ^≠ and prove that the tuples of a CCQ ^≠ query can be enumerated with a linear time precomputation and a constant delay between consecutive solutions. Moreover, under the hypothesis that the multiplication of two n×n boolean matrices cannot be done in time O(n ²), this leads to the following dichotomy for acyclic queries: either such a query is in CCQ ^≠ or it cannot be enumerated with linear precomputation and constant delay. Furthermore we prove that testing whether an acyclic formula is in CCQ ^≠ can be performed in polynomial time.

Finally, the notion of free-connex treewidth of a structure is defined. We show that for each query of free-connex treewidth bounded by some constant k, enumeration of results can be done with $O(|{\mathcal M}|^{k+1})$ precomputation steps and constant delay.