% 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.
%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.
In this work, we study the analog problem for bag semantics: computing a tuple's expected multiplicity exactly and approximately.
% Specifically, we are interested in the fine-grained complexity of computing this type of expectation based on a query result tuple's lineage polynomial which encodes how the tuple's multiplicity is computed based on the multiplicity of input tuples.
% Furthermore, we study how the complexity of this problem compares to
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.
our results imply that computing expected multiplicities for Bag-PDBs based on the results produced by such query evaluation algorithms introduces super-linear overhead (under parameterized complexity hardness assumptions/conjectures).
% 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.
%all input tuples have a fixed probability $\prob$ (s.t. $\prob \in (0,1)$).\BG{Replace with this because notion of hardness unclear: This is the case even if \ldots}
%Atri: Fair enough: droppped.
%We proceed to study how approximate multiplicities using lineage polynomials of result tuples of positive relational algebra queries ($\raPlus$) over TIDBs and for a non-trivial subclass of block-independent databases (BIDBs).
We proceed to study approximation of expected multiplicities of result tuples of positive relational algebra queries ($\raPlus$) over \AHchange{\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.
