🤖 AI Summary
This paper investigates tractability criteria for the answer counting problem #UCQ(C) of unions of conjunctive queries (UCQs). To address the impracticality of existing criteria—whose verification is infeasible for concrete UCQ classes or individual queries—we propose bounded treewidth of both the UCQ and its *contract* as a natural, verifiable parameterized tractability criterion. We prove its necessity and sufficiency within closure classes. This yields the first fixed-parameter tractability (FPT) characterization based jointly on the treewidths of the query and its contract. We further show that deciding tractability for a single UCQ is NP-hard, and even approximating the Weisfeiler–Leman dimension is computationally hard. Finally, we establish a tight complexity characterization for UCQ counting: the meta-problem is NP-hard, and existing exponential-time algorithms are optimal—thereby providing precise theoretical bounds for database query evaluation.
📝 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 Ψ ∈ C and a database D and the problem is to compute the number of answers of Ψ 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 Ψ, it is not easy to determine how hard it is to count answers to Ψ. In this work, we address the question of finding a natural tractability criterion: The combined conjunctive query of a UCQ Ψ=φ1 ∨ ... ∨ φl is the conjunctive query ^ Ψ = φ_1 ∧ ... ∧ φl. 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 Ψ, the meta problem of deciding whether #UCQ (Ψ) can be solved in time O(|D|d) is NP-hard for any fixed d ≥ 1. Moreover, we prove that a known exponential-time algorithm for solving the meta problem is optimal under assumptions from fine-grained complexity theory. As a corollary of our reduction, we also establish that approximating the Weisfeiler-Leman-Dimension of a UCQ is NP-hard.
