This problem has a practical application in probabilistic databases over multisets, where it corresponds to calculating the expected multiplicity of a query result tuple.
% It has been studied extensively for sets (lineage formulas), but the bag settings has not received much attention.
While the expectation of a polynomial can be calculated in linear time for % in the size of
polynomials % that are
in SOP form, the problem is \sharpwonehard for factorized polynomials (proven through a reduction from the problem of counting k-matchings).
%We have proven this claim through a reduction from the problem of counting k-matchings.
We prove that it is possible to approximate the expectation of a lineage polynomial in linear time
% When only considering polynomials for result tuples of
UCQs over TIDBs and BIDBs (under the assumption that there are few cancellations).
Interesting directions for future work include development of a dichotomy for bag PDBs and approximations for more general data models. % beyond what we consider in this paper.
% Furthermore, it would be interesting to see whether our approximation algorithm can be extended to support queries with negations, perhaps using circuits with monus as a representation system.
% \BG{I am not sure what interesting future work is here. Some wild guesses, if anybody agrees I'll try to flesh them out:
% \textbullet{More queries: what happens with negation can circuits with monus be used?}
% \textbullet{More databases: can we push beyond BIDBs? E.g., C-tables / aggregate semimodules or just TIDBs where each input tuple is a random variable over $\mathbb{N}$?}
% \textbullet{Other results: can we extend the work to approximate $P(R(t) = n)$}