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.
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.
