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Abstract

The class of hierarchical queries is known to define the boundary of the dichotomy between tractability and intractability for the following two extensively studied problems about self-join free Boolean conjunctive queries (SJF-BCQ): (i) evaluating a SJF-BCQ on a tuple-independent probabilistic database; (ii) computing the Shapley value of a fact in a database on which a SJF-BCQ evaluates to true. Here, we establish that hierarchical queries define also the boundary of the dichotomy between tractability and intractability for a different natural algorithmic problem, which we call the "bag-set maximization" problem. The bag-set maximization problem associated with a SJF-BCQ $Q$ asks: given a database $\cal D$, find the biggest value that $Q$ takes under bag semantics on a database $\cal D'$ obtained from $\cal D$ by adding at most $\theta$ facts from another given database $\cal D^r$. For non-hierarchical queries, we show that the bag-set maximization problem is an NP-complete optimization problem. More significantly, for hierarchical queries, we show that all three aforementioned problems (probabilistic query evaluation, Shapley value computation, and bag-set maximization) admit a single unifying polynomial-time algorithm that operates on an abstract algebraic structure, called a "2-monoid". Each of the three problems requires a different instantiation of the 2-monoid tailored for the problem at hand.