Before revision:  As a corollary, it follows that a class of conjunctive queries over graphs admits an FPRAS if  and only if it has bounded treewidth (unless BPP ) \neq \text{P})}} . In fact, our FPRAS is more general, and also applies to conjunctive queries with bounded hypertree width, as well as unions of such queries.
After revision:  As a corollary, it follows that for every class G of graphs, the counting problem for conjunctive queries whose underlying graph is in G admits an FPRAS if and only if it has bounded treewidth (unless BPP ) \neq \text{P})}} . In fact, our FPRAS is more general, and also applies to conjunctive queries with bounded hypertree width, as well as unions of such queries.