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\section{Related Work}\label{sec:related-work}

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%\subsection{Probabilistic Databases}\label{sec:prob-datab}
\textbf{Probabilistic Databases} (PDBs) have been studied predominantly for set semantics.
Approaches for probabilistic query processing (i.e., computing marginal probabilities of tuples), fall into two broad categories.
\emph{Intensional} (or \emph{grounded}) query evaluation computes the \emph{lineage} of a tuple
and then the probability of the lineage formula.
It has been shown that computing the marginal probability of a tuple is \sharpphard~\cite{valiant-79-cenrp} (by reduction from weighted model counting).
The second category, \emph{extensional} query evaluation, % avoids calculating the lineage.
% This approach
is in \ptime, but is limited to certain classes of queries.
Dalvi et al.~\cite{DS12} and Olteanu et al.~\cite{FO16} proved dichotomies for UCQs and two classes of queries with negation, respectively.
% Dalvi et al.~\cite{DS12} proved a dichotomy for UCQs:
% for any UCQ the probabilistic query evaluation problem is either \sharpphard or \ptime.
% Olteanu et al.~\cite{FO16} presented dichotomies for two classes of queries with negation.
% R\'e et al~\cite{RS09b} present a trichotomy for HAVING queries.
Amarilli et al. investigated tractable classes of databases for more complex queries~\cite{AB15}. %,AB15c
Another line of work, studies which structural properties of lineage formulas lead to tractable cases~\cite{kenig-13-nclexpdc,roy-11-f,sen-10-ronfqevpd}.
In this paper we focus on intensional query evaluation with polynomials.

Many data models have been proposed for encoding PDBs more compactly than as sets of possible worlds.
These include tuple-independent databases~\cite{VS17} (\tis), block-independent databases (\bis)~\cite{RS07}, and \emph{PC-tables}~\cite{GT06}.
%
Fink et al.~\cite{FH12} study aggregate queries over a probabilistic version of the extension of K-relations for aggregate queries proposed in~\cite{AD11d} (\emph{pvc-tables}). As an extension of K-relations, this approach supports bags. % Probabilities are computed using a decomposition approach~\cite{DBLP:conf/icde/OlteanuHK10}.
% over the symbolic expressions that are used as tuple annotations and values in pvc-tables.
% \cite{FH12} identifies a tractable class of queries involving aggregation.
In contrast, we study a less general data model ($\semNX$-PDBs)
and query class, but provide a linear time approximation algorithm and provide new insights into the complexity of computing expectations while~\cite{FH12} computes probabilities for individual output annotations.

Several techniques for approximating tuple probabilities have been proposed in related work~\cite{FH13,heuvel-19-anappdsd,DBLP:conf/icde/OlteanuHK10,DS07}, relying on Monte Carlo sampling, e.g.,~\cite{DS07}, or a branch-and-bound paradigm~\cite{DBLP:conf/icde/OlteanuHK10}.
Our approximation algorithm is also based on sampling.

\noindent \textbf{Compressed Encodings} are used for Boolean formulas (e.g, various types of circuits including OBDDs~\cite{jha-12-pdwm}) and polynomials (e.g., factorizations~\cite{factorized-db}) some of which have been utilized for  probabilistic query processing, e.g.,~\cite{jha-12-pdwm}.
Compact representations for which probabilities can be computed in linear time include OBDDs, SDDs, d-DNNF, and FBDD.
\cite{DM14c} studies circuits for absorptive semirings while~\cite{S18a} studies circuits that include negation (expressed as the monus operation). Algebraic Decision Diagrams~\cite{bahar-93-al} (ADDs) generalize BDDs to variables with more than two values. Chen et al.~\cite{chen-10-cswssr} introduced the generalized disjunctive normal form.
%\noindent
\Cref{sec:param-compl} covers more related work on fine-grained complexity.


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