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Conjunctive queries are basic and heavily studied database queries; inrelational algebra, they are the select-project-join queries. In this article,we study the fundamental problem of counting, given a conjunctive query and arelational database, the number of answers to the query on the database. Inparticular, we study the complexity of this problem relative to sets ofconjunctive queries. We present a trichotomy theorem, which shows essentiallythat this problem on a set of conjunctive queries is either tractable,equivalent to the parameterized CLIQUE problem, or as hard as the parameterizedcounting CLIQUE problem; the criteria describing which of these situationsoccurs is simply stated, in terms of graph-theoretic conditions.

A Trichotomy in the Complexity of Counting Answers to Conjunctive Queries