Conjunctive Queries over Trees

TL;DR

This paper analyzes the complexity and expressive power of conjunctive queries over unranked labeled trees, establishing a clear boundary between tractable and NP-complete cases and exploring query expressiveness.

Contribution

It provides a comprehensive framework for evaluating the complexity of conjunctive queries over trees and characterizes the tractability frontier with a dichotomy theorem.

Findings

Polynomial-time evaluation for certain axis sets

NP-completeness for other axis sets

Every conjunctive query has an equivalent acyclic positive query

Abstract

We study the complexity and expressive power of conjunctive queries over unranked labeled trees represented using a variety of structure relations such as ``child'', ``descendant'', and ``following'' as well as unary relations for node labels. We establish a framework for characterizing structures representing trees for which conjunctive queries can be evaluated efficiently. Then we completely chart the tractability frontier of the problem and establish a dichotomy theorem for our axis relations, i.e., we find all subset-maximal sets of axes for which query evaluation is in polynomial time and show that for all other cases, query evaluation is NP-complete. All polynomial-time results are obtained immediately using the proof techniques from our framework. Finally, we study the expressiveness of conjunctive queries over trees and show that for each conjunctive query, there is an equivalent acyclic positive query (i.e., a set of acyclic conjunctive queries), but that in general this query is not of polynomial size.