Getting to the Root: A Combined Complexity Perspective on Consistent Query Answering

Abstract

We study the combined complexity of consistent query answering for Boolean self-join-free conjunctive queries with unary primary keys and acyclic attack graphs. Although every fixed query in this class admits a first-order rewriting [21], we show that allowing the query to vary makes the problem Pi_2^P-complete. To isolate the query structure governing this transition, we introduce the closure generator size: the minimum number of query variables whose closure under the functional dependencies induced by the primary keys contains every query variable. Our main theorem shows that every fixed bound on this parameter yields polynomial-time combined complexity. Parameterized by the closure generator size, the problem is in XP and co-W[t]-hard for every fixed t. The polynomial-time result is specific to unary primary keys: with binary primary keys, the problem becomes coNP-hard already for self-join-free queries with acyclic attack graphs and closure generator size zero. Regarding the upper bound, CQA is in coNP for arbitrary Boolean conjunctive queries of bounded closure treewidth or bounded hyperclosure treewidth; these width measures originate in [1]. Our polynomial-time algorithm realizes quantifier alternation dynamically by interleaving existential and universal substitutions with variable-level reductions. To implement this approach, we introduce concepts and techniques such as attack-propagation graphs, variable guarding and guarded pruning, database and query saturation, and query normalization.