Counting Answers to Unions of Conjunctive Queries: Natural Tractability Criteria and Meta-Complexity

TL;DR

This paper establishes a natural criterion based on treewidth for determining when counting answers to unions of conjunctive queries is fixed-parameter tractable, simplifying previous complex criteria and providing complexity insights.

Contribution

It introduces a natural tractability criterion based on combined conjunctive queries and their contracts, linked to bounded treewidth, for counting answers to UCQs.

Findings

Fixed-parameter tractability characterized by bounded treewidth of combined queries.

Closure properties are necessary for a natural tractability criterion.

Deciding tractability for a single UCQ is NP-hard for fixed parameters.

Abstract

We study the problem of counting answers to unions of conjunctive queries (UCQs) under structural restrictions on the input query. Concretely, given a class C of UCQs, the problem #UCQ(C) provides as input a UCQ Q in C and a database D and the problem is to compute the number of answers of Q in D. Chen and Mengel [PODS'16] have shown that for any recursively enumerable class C, the problem #UCQ(C) is either fixed-parameter tractable or hard for one of the parameterised complexity classes W[1] or #W[1]. However, their tractability criterion is unwieldy in the sense that, given any concrete class C of UCQs, it is not easy to determine how hard it is to count answers to queries in C. Moreover, given a single specific UCQ Q, it is not easy to determine how hard it is to count answers to Q. In this work, we address the question of finding a natural tractability criterion: The combined conjunctive query of a UCQ $\varphi_1 \vee \dots \vee \varphi_\ell$ is the conjunctive query $\varphi_1 \wedge \dots \wedge \varphi_\ell$. We show that under natural closure properties of C, the problem #UCQ(C) is fixed-parameter tractable if and only if the combined conjunctive queries of UCQs in C, and their contracts, have bounded treewidth. A contract of a conjunctive query is an augmented structure, taking into account how the quantified variables are connected to the free variables. If all variables are free, then a conjunctive query is equal to its contract; in this special case the criterion for fixed-parameter tractability of #UCQ(C) thus simplifies to the combined queries having bounded treewidth. Finally, we give evidence that a closure property on C is necessary for obtaining a natural tractability criterion: We show that even for a single UCQ Q, the meta problem of deciding whether #UCQ({Q}) can be solved in time $O(|D|^d)$ is NP-hard for any fixed $d\geq 1$.