Structural Tractability of Counting of Solutions to Conjunctive Queries

We explore the complexity of counting solutions to conjunctive queries, a basic class of queries from database theory. We introduce a parameter, called the quantified star size of a query ϕ, which measures how the free variables are spread in ϕ. As usual in database theory, we associate a hypergraph to a query ϕ. We show that for classes of queries for which these associated hypergraphs admit good decompositions, e.g., bounded width generalized hypertree decompositions, bounded quantified star size exactly characterizes the subclasses of hypergraphs for which counting the number of solutions is tractable. In the case of bounded arity, this allows us to fully characterize the classes of hypergraphs for which counting the solutions is tractable. Finally, we also analyze the complexity of computing the quantified star size of a conjunctive query.

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