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\begin{abstract}
%  The problem of computing the marginal probability of a tuple in the result of a query over set-probabilistic databases (PDBs) can be reduced to calculating the probability of the \emph{lineage formula} of the result, a Boolean formula over random variables representing the existence of tuples in the database's possible worlds.
  The problem of computing the marginal probability of a tuple in the result of a query over set-probabilistic databases (PDBs) is arguably the most fundamental problem in set-PDBs.
%can be reduced to calculating the probability of the \emph{lineage formula} of the result, a Boolean formula over random variables representing the existence of tuples in the database's possible worlds.
  %The analog for bag semantics is a natural number-valued polynomial over random variables that evaluates to the multiplicity of the tuple in each world.
  The analog for bag semantics is computing the expected multiplicity of a result tuple.
  %In this work, we study the problem of calculating the expectation of such polynomials (a tuple's expected  multiplicity) exactly and approximately.
  In this work, we study the problem of a tuple's expected  multiplicity exactly and approximately.
  We are specifically interested in the fine-grained complexity of this problem relative to the complexity of deterministic query evaluation --- if these complexities are comparable, it opens the door to practical deployment of probabilistic databases.
  Unfortunately, we show the reverse; our results imply that computing expected multiplicities for Bag-PDB based on the results produced by such algorithms introduces super-linear overhead.
  % Such factorized representations are necessary to realize the performance of modern join algorithms (e.g., worst-case optimal joins), and so our results imply that a Bag-PDB doing exact computations (via these factorized representations) can never be as fast as a classical (deterministic) database.
  The problem stays hard even if all input tuples have a fixed probability $\prob$ (s.t. $\prob \in (0,1)$).
  We proceed to study polynomials of result tuples of positive relational algebra queries ($\raPlus$) over TIDBs and for a non-trivial subclass of block-independent databases (BIDBs).
  We develop a sampling algorithm that computes a $1 \pm \epsilon$-approximation of the expected multiplicity of an output tuple in linear time in the runtime of a comparable deterministic query.
  % By removing Bag-PDB's reliance on the sum-of-products representation of polynomials, this result paves the way for future work on PDBs that are competitive with deterministic databases.
\end{abstract}

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