Counting Answers to Unions of Conjunctive Queries: Natural Tractability Criteria and Meta-Complexity
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2023
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| _version_ | 1866914340173512704 |
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| author | Focke, Jacob Goldberg, Leslie Ann Roth, Marc Živný, Stanislav |
| author_facet | Focke, Jacob Goldberg, Leslie Ann Roth, Marc Živný, Stanislav |
| contents | 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 $φ_1 \vee \dots \vee φ_\ell$ is the conjunctive query $φ_1 \wedge \dots \wedge φ_\ell$. 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 $d\geq 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_10634 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Counting Answers to Unions of Conjunctive Queries: Natural Tractability Criteria and Meta-Complexity Focke, Jacob Goldberg, Leslie Ann Roth, Marc Živný, Stanislav Discrete Mathematics Databases 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 $φ_1 \vee \dots \vee φ_\ell$ is the conjunctive query $φ_1 \wedge \dots \wedge φ_\ell$. 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 $d\geq 1$. |
| title | Counting Answers to Unions of Conjunctive Queries: Natural Tractability Criteria and Meta-Complexity |
| topic | Discrete Mathematics Databases |
| url | https://arxiv.org/abs/2311.10634 |