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\begin{abstract}
In this work, we study the problem computing a tuple's expected multiplicity over bag-\abbrTIDB\xplural exactly and approximately.
  We consider bag-\abbrTIDB\xplural where we have a bound $\bound$ on the maximum multiplicity (we refer to such bag-\abbrTIDB\xplural as \abbrCTIDB\xplural). In this work we consider the case when $\bound$ is a constant (since that is what is used in practice). We are specifically
   interested in the fine-grained complexity and how it compares to the complexity of deterministic query evaluation algorithms --- 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 \abbrCTIDB\xplural based on the results produced by such query evaluation algorithms introduces super-linear overhead (under parameterized complexity hardness assumptions/conjectures).
  We proceed to study approximation of expected multiplicities of result tuples of positive relational algebra queries ($\raPlus$) over \abbrCTIDB\xplural and for a non-trivial subclass of block-independent databases (\abbrBIDB\xplural).
  We develop a sampling algorithm that computes a $(1 \pm \epsilon)$-approximation of the expected multiplicity of an output tuple in time linear in the runtime of a comparable deterministic query for any $\raPlus$ 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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