A Dichotomy for the Generalized Model Counting Problem for Unions of Conjunctive Queries

We study the $generalized~model~counting~problem$, defined as follows: given a database, and a set of deterministic tuples, count the number of subsets of the database that include all deterministic tuples and satisfy the query. This problem is computationally equivalent to the evaluation of the query over a tuple-independent probabilistic database where all tuples have probabilities in $\{0,\frac{1}{2},1\}$. Previous work has established a dichotomy for Unions of Conjunctive Queries (UCQ) when the probabilities are arbitrary rational numbers, showing that, for each query, its complexity is either in polynomial time or #P-hard. The query is called $safe$ in the first case, and $unsafe$ in the second case. Here, we strengthen the hardness proof, by proving that an unsafe UCQ query remains #P-hard even if the probabilities are restricted to $\{0,\frac{1}{2},1\}$. This requires a complete redesign of the hardness proof, using new techniques. A related problem is the $model~counting~problem$, which asks for the probability of the query when the input probabilities are restricted to $\{0,\frac{1}{2}\}$. While our result does not extend to model counting for all unsafe UCQs, we prove that model counting is #P-hard for a class of unsafe queries called Type-I forbidden queries.