The Complexity of Weighted Counting for Acyclic Conjunctive Queries
  This paper is a study of weighted counting of the solutions of acyclic conjunctive queries ($\ACQ$). The unweighted quantifier free version of this problem is known to be tractable (for combined complexity), but it is also known that introducing even a single quantified variable makes it $\sP$-hard. We first show that weighted counting for quantifier-free $\ACQ$ is still tractable and that even minimalistic extensions of the problem lead to hard cases. We then introduce a new parameter for quantified queries that permits to isolate large island of tractability. We show that, up to a standard assumption from parameterized complexity, this parameter fully characterizes tractable subclasses for counting weighted solutions of $\ACQ$ queries. Thus we completely determine the tractability frontier for weighted counting for $\ACQ$. 
Arnaud Durand and Stefan Mengel
cs.CC cs.LO math.LO