%!TEX root=./main.tex
%root: main.tex
\section{Introduction}\label{sec:intro}

\secrev{
This work explores the problem of computing the expectation of a tuple's multiplicity in a bag \abbrTIDB.
Our analysis specifically considers a restricted form of bag \abbrTIDB which we call a \abbrCTIDB.  A \abbrCTIDB,
$\pdb = \inparen{\inset{0,\ldots, c}^\numvar, \mathcal{P}}$ is a bag of tuples such that each tuple models $c$ disjoint events, where each such set of disjoint events is itself independent of the others.  A tuple in $\pdb$ has a multiplicity of at most $c$.  The element $\inset{0,\ldots,c }^\numvar$ is the set of all worlds and $\mathcal{P}$ is the product distribution over the set of all worlds.  A given world $\vct{W} = \inset{0,\ldots, c}$ can be interpreted for each $\vct{W}\pbox{i}= j$ as denoting that tuple $\tup_i$ appears $j$ times in world $\vct{W}$ for $j \in [0, c]$.  The resulting product distribution can then be expressed across the $\numvar$ base tuples of the encoding as $\prob_{i, j} = \probOf\pbox{W\pbox{i} = j}$, where each distribution is independent for $i \in [\numvar]$.
}  
%\mypar{For a later section}
%\sout{
%Since each tuple in $\pdb$ has a mutually exclusive probability distribution over its possible multiplicities, it is natural to reduce a \abbrCTIDB to traditional (set) block independent database (\abbrBIDB).  We refer to the reduced \abbrBIDB as a $1$-\abbrBIDB, as it is the case that each tuple can appear in a possible world at most $c = 1$ time.  \Cref{fig:ctidb-red} shows an example of this reduction.
%}  
\secrev{
Allowing for $\leq c$ multiplicities across all tuples gives rise to having $\leq \inparen{c+1}^\numvar$ possible worlds instead of the usual $2^\numvar$ possible worlds of a (set) $1$-\abbrTIDB. 
In this work, it is natural to be specifically considering bag query semantics.

We can formally state this problem as:

\begin{Problem}\label{prob:expect-mult}
Given a \abbrCTIDB $\pdb = \inparen{\idb, \pd}$, query $\query$, and result tuple $\tup$, compute the expected multiplicity of $\tup$: $\expct_{\randDB\sim\pd}\pbox{\query\inparen{\randDB}\inparen{\tup}}$.
\end{Problem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Example of product distribution of c-TIDB
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\begin{figure}[h!]
%	\centering
%	\textcolor{red}{
%	\begin{tabular}{>{\footnotesize}c | >{\footnotesize}c | >{\footnotesize}c | >{\footnotesize}c | >{\footnotesize}c}
%		\multicolumn{5}{c}{$\mathbf{\rel}$}\\
%		\toprule
%		A & Mult. $\inparen{M}$ &$\probOf\pbox{M=1}$ &$\probOf\pbox{M=2}$ &$\probOf\pbox{M=3}$\\
%		\midrule
%		$a$ & $2$ & $0.4$ &$0.3$ &$0.0$\\
%		$b$ & $3$ & $0.2$ &$0.35$ &$0.15$\\
%	\end{tabular}
%%	\hspace*{0.5cm}
%%	{\LARGE $\Rightarrow$}
%%	\hspace*{0.5cm}
%	\begin{tabular}{>{\footnotesize}c | >{\footnotesize}c}
%		\multicolumn{2}{c}{$\textbf{World Probabilities}$}\\
%		\toprule
%		World & Probability \\
%		\midrule
%		$\emptyset$ & $0.3\cdot0.3 = 0.09$\\
%		$\inset{\intup{a, 1}}$ & $0.4\cdot0.3 = 0.12$\\
%		$\inset{\intup{a, 2}}$ & $0.3\cdot0.3 = 0.09$\\
%		$\inset{\intup{b, 1}}$ & $0.3\cdot0.2 = 0.06$\\
%		$\inset{\intup{b, 2}}$ & $0.3\cdot0.35 = 0.105$\\
%		$\inset{\intup{b, 3}}$ & $0.3\cdot0.15 = 0.045$\\
%		$\inset{\intup{a, 1}, \intup{b, 1}}$ & $0.4\cdot0.2 = 0.08$\\
%		$\inset{\intup{a, 1}, \intup{b, 2}}$ & $0.4\cdot0.35 = 0.14$\\
%		$\inset{\intup{a, 1}, \intup{b, 3}}$ & $0.4\cdot0.15 = 0.06$\\
%		$\inset{\intup{a, 2}, \intup{b, 1}}$ & $0.3\cdot0.2 = 0.06$\\
%		$\inset{\intup{a, 2}, \intup{b, 2}}$ & $0.3\cdot0.35 = 0.105$\\
%		$\inset{\intup{a, 2}, \intup{b, 3}}$ & $0.3\cdot0.15 = 0.045$\\
%	\end{tabular}
%	}
%	\caption{\textcolor{red}{\abbrCTIDB relation$\rel$ and its possible worlds with their probabilities.% Reduction to $1$-\abbrBIDB ($\rel'$).  Note the probability distribution over tuple multiplicities of $\rel$, where e.g. tuple $\tup_1$ has a probability $\prob_{1, j} > 0$ for each multiplicity $j \in [2]$.  This is better expressed as a block of mutually exclusive tuples, where each tuple has a specific multiplicity in $[c]$.  Multiplicities that don't exist are automatically assigned a probability of $0$.  Also note that it is implicit in \abbrBIDB\xplural for a block $i$ that $1 - \sum_{j \in [c]}\prob_{i, j}$ is the probability that no tuple in that block will be selected for a possible world.
%}}
%	\label{fig:ctidb-red}
%\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Figure may work for an example in a later section of the Intro
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[h!]
%	\centering
%	\textcolor{red}{
%	\begin{tabular}{c | c | c}
%		\multicolumn{3}{c}{$\mathbf{\rel}$}\\
%		\toprule
%		A & Mult. & id\\
%		\midrule
%		$a$ & $2$ & $\tup_1$\\
%		$b$ & $3$ & $\tup_2$\\
%	\end{tabular}
%	\hspace*{1cm}
%	{\LARGE $\equiv$}
%	\hspace*{1cm}
%	\begin{tabular}{c | c | c}
%		\multicolumn{3}{c}{$\textbf{\rel'}$}\\
%		\toprule
%		A & $\poly$ & $\probOf\pbox{x_{i, j}}$\\
%		\midrule
%		a & $\pVar_{1, 1}$ & $0.4$\\
%		a & $\pVar_{1, 2}$ & $0.3$\\
%		a & $\pVar_{1, 3}$ & $0.0$\\
%		\midrule
%		b & $\pVar_{2, 1}$ & $0.2$\\
%		b & $\pVar_{2, 2}$ & $0.35$\\
%		b & $\pVar_{2, 3}$ & $0.15$\\
%	\end{tabular}
%	}
%	\caption{\textcolor{red}{\abbrCTIDB ($\rel$) Reduction to $1$-\abbrBIDB ($\rel'$).  Note the probability distribution over tuple multiplicities of $\rel$, where e.g. tuple $\tup_1$ has a probability $\prob_{1, j} > 0$ for each multiplicity $j \in [2]$.  This is better expressed as a block of mutually exclusive tuples, where each tuple has a specific multiplicity in $[c]$.  Multiplicities that don't exist are automatically assigned a probability of $0$.  Also note that it is implicit in \abbrBIDB\xplural for a block $i$ that $1 - \sum_{j \in [c]}\prob_{i, j}$ is the probability that no tuple in that block will be selected for a possible world.
%}}
%	\label{fig:ctidb-red}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We upperbound the multiplicity of tuples in a \abbrCTIDB since this is what typically seen in practice.
%because of the cancellation effect of queries over a $1$-\abbrBIDB (introduced later), where, for the worst case, a self join query, we would have a factor of $\frac{1}{c^{n-1}}$ cancellations. 
Allowing for unbounded $c$ is an interesting open problem.
}

%\sout{
%\mypar{Example that can perhaps be used later on (using commented out figure above)}
%Given a \abbrCTIDB $\pdb$ with $\numvar$ tuples, we can encode a possible world by the vector $\vct{W} \in \inset{0,\ldots, c}^\numvar$, with the intuitive interpretation when bit $W_i = j$, then tuple $\tup_i$ with multiplicity $j$ is selected, with $\tup_i$ not existing for the special case of $j = 0$.  For the example in ~\cref{fig:ctidb-red}, we have that for \abbrCTIDB $\textbf{R}$, $\numvar = 2$.  Then, e.g., arbitrary world vector $\vct{W} = [2, 3]$ encodes the possible world $\db = \inset{\intup{a, 2}, \intup{b, 3}}$  Computing ~\cref{prob:expect-mult} for tuple $\tup_2$ in ~\cref{fig:ctidb-red} when $\query = \mathbf{\rel}$ then becomes $\expct_{\randDB\sim\pd}\pbox{\mathbf{\rel}\inparen{\tup_2}} = 1\cdot\prob_{2,1} + 2\cdot\prob_{2,2} + 3\cdot\prob_{2,3} = 1\cdot 0.2 + 2\cdot 0.35 + 3\cdot 0.15 = 1.35$.
%}

\AR{I know we did not decide on this ealier but I think this would be a good place have a para on results on setPDBs and focus just on $1$-TIDBs. Mention with set query eval semantics the problem is really and summarize at least the lower bound result here. Also might be a good place to mention the `trivial' algo for \abbrCTIDB in a sentence or so: the idea is to establish that we can solve our problem in polytime. So the interesting Q is fine-grained/parameterized complexity.}

\secrev{
One of the main theoretical points in this work is to discern whether or not bag \abbrPDB query semantics are indeed linear in the runtime of the deterministic query.
\AR{Again just mention \abbrCTIDB instead of general \abbrPDB. Also you should motivate in a sentence or so WHY this is a problem to consider. Again if we have already motivated \abbrCTIDB because of practical considerations then you can say here than a positive answer would mean these things nice things in practice.}
Unfortunately, we prove that this is not the case.  To analyze this question we denote by $\timeOf{}^*(Q,\pdb)$ the optimal runtime complexity of computing ~\cref{prob:expect-mult} over \abbrCTIDB $\pdb$.  Let $\qruntime{\query, \db}$ be the runtime of query $\query$ on deterministic database $\db$ under a cost model that is satisfied by a wide range of query processing algorithms, including those based on the recent work on worst-case optimal join algorithms.
\AR{First I though we were going with $T_{det}^*$ instead of just $\qruntime{\cdot}$-- this change needs to be made for the rest of the intro. Also for now you just informally define $T_{det}^*$ as the {\em optimal} deterministic runutime (but add a parenthetical remark on there being some caveats). Second, $\db$ is not connected to $\pdb$.}
We make this runtime concrete later on.
We denote by $\dbbase$ the base \abbrCTIDB table containing all possible tuples, formally as,
\begin{Definition}[$\dbbase$]
Let $\dbbase$ be the relation composed of all possible tuples in $\pdb$, i.e. $\dbbase = \bigcup_{\db \in \idb}\db$.
\end{Definition}
\AR{Again if we are defining \abbrCTIDB `from scratch' instead of in terms of general PDBs, then the above might not be needed. Also it should be \abbrCTIDB instead of \abbrPDB in the sentence below.}
~\Cref{tab:lbs} shows our lower bounds for computing ~\cref{prob:expect-mult} on general \abbrPDB\xplural.  
\newline 
\begin{table}[h!]
\begin{tabular}{|p{0.43\textwidth}|p{0.12\textwidth}|p{0.35\textwidth}|}
\hline
Lower bound on $\timeOf{}^*(\query,\pdb)$ & Num. $\pd$s & Hardness Assumption\\
\hline
$\Omega\inparen{\inparen{\qruntime{\query, \dbbase}}^{1+\eps_0}}$ for {\em some} $\eps_0>0$ & Single & Triangle Detection hypothesis\\
%\hline
$\omega\inparen{\inparen{\qruntime{\query, \dbbase}}^{C_0}}$ for {\em all} $C_0>0$ & Multiple &$\sharpwzero\ne\sharpwone$\\
%\hline
$\Omega\inparen{\inparen{\qruntime{\query, \dbbase}}^{c_0\cdot k}}$ for {\em some} $c_0>0$ & Multiple & \Cref{conj:known-algo-kmatch}\\ %Multiple & Current $k$-matching algorithms\\
\hline
\end{tabular}
\caption{Our lower bounds for a specific hard query $Q$ parameterized by $k$. The $\pdb$ is over the same (family of) $\dbbase$ and those with `Multiple' in the second column need the algorithm to be able to handle multiple $\pd$ (for a given $\dbbase$). The last column states the hardness assumptions that imply the lower bounds in the first column ($\eps_o,C_0,c_0$ are constants that are independent of $k$).}
\label{tab:lbs}
\end{table}
\mypar{Our lower bound results} In table~\ref{tab:lbs} we show that depending on what hardness result/conjecture we assume, we get various emphatic versions of {\em no} as an answer to our question.  To make some sense of the other lower bounds in Table~\ref{tab:lbs}, we note that it is not too hard to show that $\timeOf{}^*(Q,\pdb) \le  O\inparen{\inparen{\qruntime{Q, \dbbase}}^k}$, where $k$ is the largest degree of the polynomial $\apolyqdt$ over all result tuples $\tup$ (and the parameter that defines our family of hard queries).
\AR{$\Phi$ is no longer defined so the above needs to be re-written just in terms if the query.}
What our lower bound in the third row says is that one cannot get more than a polynomial improvement over essentially the trivial algorithm for \Cref{prob:informal}.\AH{Not sure what is meant by `the trivial algorithm for (what was originally called) Problem 1.4'}
\AR{Trivial algorithm us convert to SMB form and then use linearity of expectation.}
 However, this result assumes a hardness conjecture that is not as well studied as those in the first two rows of the table (see \Cref{sec:hard} for more discussion on the hardness assumptions). Further, we note that existing results already imply the claimed lower bounds if we were to replace the $\qruntime{\query, \dbbase}$ by just $\abs{\dbbase}$ (indeed these results follow from known lower bound for deterministic query processing). Our contribution is to then identify a family of hard queries where deterministic query processing is `easy' but computing the expected multiplicities is hard. 

\mypar{Our upper bound results} We introduce an $(1\pm \epsilon)$-approximation algorithm that computes ~\cref{prob:expect-mult} in $\leq \qruntime{\query, \dbbase}$. In particular, we show the following upper bound results.
(i) We show that e.g. for a circuit representation of the lineage polynomial (more on this later), when the circuit is a tree and there is a single
 result tuple, we also have the same runtime  (we can also handle the case of multiple result tuples\footnote{We can approximate the expected result tuple multiplicities (for all result tuples {\em simultanesouly}) with only $O(\log{Z})=O_k(\log{n})$ overhead (where $Z$ is the number of result tuples) over the runtime of a broad class of query processing algorithms (see \Cref{app:sec-cicuits}).}).
Further, we show that for {\em any} $\raPlus$ query on a \abbrTIDB $(1$-$\abbrTIDB)$, we also obtain linear runtime for approximation.
% the approximation algorithm has runtime linear in the size of the compressed lineage encoding (
In contrast, known approximation techniques (\cite{DBLP:conf/icde/OlteanuHK10,DBLP:journals/jal/KarpLM89}) in set-\abbrPDB\xplural need time $\Omega(\abs{\circuit}^{2k})$ (see \Cref{sec:karp-luby}).
\AR{The above needs to be re-written. It is stated in terms of circuits but circuits are only part of our solution. You should state the final result: i.e. we have a runtime of $O_{\eps}(T_{det}^*)$.}
(ii) We generalize the \abbrPDB data model considered by the approximation algorithm to a class of bag-Block Independent Disjoint Databases (see \Cref{subsec:tidbs-and-bidbs}) (\abbrBIDB\xplural).
}
\secrev{
\subsection{Polynomial Equivalence}
A common encoding of probabilistic databases (e.g., in \cite{IL84a,Imielinski1989IncompleteII,Antova_fastand,DBLP:conf/vldb/AgrawalBSHNSW06} and many others) relies on annotating tuples with lineages, propositional formulas that describe the set of possible worlds that the tuple appears in.  The bag semantics analog is a provenance/lineage polynomial $\apolyqdt$~\cite{DBLP:conf/pods/GreenKT07} (see~\Cref{fig:nxDBSemantics} for a definition), a polynomial with non-zero integer coefficients and exponents, over integer variables $\vct{X}$ encoding input tuple multiplicities.
\begin{figure}
  \begin{align*}
	  \polyqdt{\project_A(\query)}{\dbbase}{\tup} =& \sum_{\tup': \project_A(\tup') = \tup} \polyqdt{\query}{\dbbase}{\tup'} &
	  \polyqdt{\query_1 \union \query_2}{\dbbase}{\tup} =& \polyqdt{\query_1}{\dbbase}{\tup} + \polyqdt{\query_2}{\dbbase}{\tup}\\
	  \polyqdt{\select_\theta(\query)}{\dbbase}{\tup} =& \begin{cases}
	    \polyqdt{\query}{\dbbase}{\tup} & \text{if }\theta(\tup) \\
	    0                       & \text{otherwise}.
	    \end{cases} &
	       \begin{aligned}
	          \polyqdt{\query_1 \join \query_2}{\dbbase}{\tup} =\\ ~
	        \end{aligned}&
	          \begin{aligned}
	            &\polyqdt{\query_1}{\dbbase}{\project_{\attr{\query_1}}{\tup}}  \\
	            &~~~\cdot\polyqdt{\query_2}{\dbbase}{\project_{\attr{\query_2}}{\tup}}
	          \end{aligned}\\
	                                           & & & \polyqdt{\rel}{\dbbase}{\tup} = \sum_{j \in [c]}j\cdot\pVar_{\tup, j}
	\end{align*}\\[-10mm]
	\caption{Construction of the lineage (polynomial) for an $\raPlus$ query over a \abbrBPDB, where $\vct{X}$ consists of all $X_\tup$ over all $\rel$ in $\dbbase$ and $\tup$ in $\rel$. Here $\dbbase.\rel$ denotes the instance of relation $\rel$ in $\dbbase$.} 
	\label{fig:nxDBSemantics}
\end{figure}

We drop $\query$, $\dbbase$, and $\tup$ from $\apolyqdt$ when they are clear from the context or irrelevant to the discussion. We now specify the problem of computing the expectation of tuple multiplicity in the language of lineage polynomials:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Problem}[Expected Multiplicity of Lineage Polynomials]\label{prob:bag-pdb-poly-expected}
Given an $\raPlus$ query $\query$, 
\AHchange{
\abbrCTIDB $\pdb$
}
and result tuple $\tup$, compute the expected
multiplicity of the polynomial $\apolyqdt$ (i.e., $\expct_{\vct{W}\sim \pdassign}\pbox{\apolyqdt(\vct{W})}$).,
where $\pdassign$ is the distribution induced by $\pd$ on the relevant assignments $\vct{W}$ to variables of $\apolyqdt$.
\end{Problem}
We note that computing \Cref{prob:expect-mult} 
is equivalent to computing \Cref{prob:bag-pdb-poly-expected} (see \Cref{prop:expection-of-polynom}).
In this work, we study the complexity of \Cref{prob:bag-pdb-poly-expected} for several models of probabilistic databases and various encodings of such polynomials.
}

A probabilistic database (PDB) $\pdb$ is a pair $\inparen{\idb, \pd}$, where $\idb$ is a set of deterministic database instances called possible worlds and $\pd$ is a probability distribution over $\idb$.
\AHchange{
A tuple independent database (\abbrTIDB) (to which we will refer to later) is a \abbrPDB such that each tuple is an independent random event.
}
A commonly studied problem in probabilistic databases is, given a query $\query$, PDB $\pdb$, and possible query result tuple $\tup$, to compute the tuple's \textit{marginal probability} of being in the query's result, i.e., computing the expectation of a Boolean  random variable over $\pd$ that is $1$ for every $\db \in \idb$ for which $\tup \in \query(\db)$ and $0$ otherwise.
In this work, we are interested in bag semantics, where each tuple is associated with a multiplicity.
Following~\cite{DBLP:conf/pods/GreenKT07}, we model bag databases (resp., relations) as functions from each $\tup$ to the tuple's multiplicity $\db(\tup) \in \semN$ in a possible world $\db$.
\sout{
We refer to such a probabilistic database as a bag-probabilistic database or \abbrBPDB for short.
}

The natural generalization of the \AHchange{(set)} problem of computing marginal probabilities of query result tuples to bag semantics is to compute the expectation of a random variable over $\pd$ that is assigned value $\query(\db)(\tup) \in \semN$ in world $\db \in \idb$
\AHchange{
, formally $\expct_{\randDB\sim\pd}\pbox{\query\inparen{\randDB}\inparen{\tup}}$.
}

For \lstinline{COUNT(*)} queries, expected multiplicities can model the expected count.  \sout{
The equivalent set-\abbrPDB operation, simply computes the probability that this count is non-zero.
Further,
} We are interested in the parameterized complexity of 
\AHchange{
computing the expectation,
}%\Cref{prob:bag-pdb-query-eval} 
(i.e. we think of $\query$ as being parameterized by some parameter $k$ with the size of the database going to infinity relative to $k$). Unless stated otherwise, we implicitly assume the probability distribution $\pd$, and for notational convenience use $\expct\pbox{\cdot}$ instead of $\expct_\pd\pbox{\cdot}$. 

\AHchange{
While the parameterized and fine-grained results of this paper apply to general \abbrPDB\xplural, we start by focusing on a restricted form of \abbrTIDB which we refer to as \abbrCTIDB.
As alluded to, a \abbrTIDB is a compressed encoding of probabilistic databases where the presence of each individual tuple (out of a total of $\numvar$ input tuples) in a possible world is modeled as an independent probabilistic event.\footnote{
  This model is exactly the definition of \abbrTIDB{}s \cite{VS17} under set semantics. Note that this is only one possible definition of \abbrTIDB{}s under bag semantics. In \Cref{sec:gener-results-beyond} we discuss alternatives and to what degree our results extend to these alternatives.\label{footnote:set-not-limit}
  }
We will denote $\dbbase=(t_1,\dots,t_\numvar)$. Each of the $2^n$ possible worlds in $\Omega$ can be encoded as a string in $\{0,1\}^\numvar$. In particular, any vector $\vct{W}=\inparen{W_1,\dots,W_n}\in \{0,1\}^\numvar$ represents a world $\db\in\idb$ in the natural way: i.e. $\tup_i\in\db$
iff $W_i=1$. Furthermore, $\pd$ is compactly described by a tuple $\vct{p}=\inparen{p_1,\dots,p_n}$, which induces the Bernoulli distribution over vectors $\vct{W}\in\{0,1\}^\numvar$ where each $i\in [n]$, $\probOf(W_i=1)=p_i$.
We then define a \abbrCTIDB be a bag \abbrTIDB with the further restriction that each tuple $\tup$ has a multiplicity of at most some constant $c$, formally: $\forall \db \in \pdb, ~\forall \tup \in \db, ~\db\inparen{\tup}\leq c$.  That is, any tuple in a \abbrCTIDB has a multiplicity of at most $c$.
}


%\mypar{\abbrTIDB\xplural}
%We initially focus on tuple-independent probabilistic bag-databases\footnote{See \cite{DBLP:series/synthesis/2011Suciu} for a survey of set-\abbrTIDBs; the bag encoding is analogous~\cite{DBLP:conf/pods/GreenKT07}.} (\abbrTIDB\xplural), a compressed encoding of probabilistic databases where the presence of each individual tuple (out of a total of $\numvar$ input tuples) in a possible world is modeled as an independent probabilistic event.\footnote{
%  This model is exactly the definition of \abbrTIDB{}s \cite{VS17} under set semantics. Note that this is only one possible definition of \abbrTIDB{}s under bag semantics. In \Cref{sec:gener-results-beyond} we discuss alternatives and to what degree our results extend to these alternatives.\label{footnote:set-not-limit}
%  % Mirroring the implementation of bag relations in production database systems (e.g., Postgresql, DB2), tuple multiplicities are modeled by retaining copies of each tuple (up to its largest possible multiplicity).
%  % % To make each duplicate tuple unique in a set-\abbrTIDB we can assign unique keys across all duplicates.
%  % When the multiplicity of input tuple is bound by some constant,
%  % the increased input size is negligible.\label{footnote:set-not-limit}
%}
%% OK: I tidied things up a touch.
%%\BG{The footnote is still a bit hard to follow I think, but I do not have a great suggestion on how to improve it.}
%We will denote $\dbbase=(t_1,\dots,t_\numvar)$. Each of the $2^n$ possible worlds in $\Omega$ can be encoded as a string in $\{0,1\}^\numvar$. In particular, any vector $\vct{W}=\inparen{W_1,\dots,W_n}\in \{0,1\}^\numvar$ represents a world $\db\in\idb$ in the natural way: i.e. $\tup_i\in\db$
%iff $W_i=1$. Furthermore, $\pd$ is compactly described by a tuple $\vct{p}=\inparen{p_1,\dots,p_n}$, which induces the Bernoulli distribution over vectors $\vct{W}\in\{0,1\}^\numvar$ where each $i\in [n]$, $\probOf(W_i=1)=p_i$.

%Finally for each $\vct{W}\in\{0,1\}^\numvar$, we define $\pdb_{\vct{W}}$
%\AH{Where do we use this notation?  If we use this somewhere, should we maybe use $\db_{\vct{\randWorld}}$ instead?}
% as the world represented by $\vct{W}$.
%Atri: I don't think we use it so removing it.

%Atri: Stuff below was confusing, so am re-writing it.
%A \abbrTIDB encodes a compatible $\pdb$ as a deterministic database $\encodedDB$ with $\numvar$ tuples, each annotated with a probability $\prob_\tup$, and with $\pd$
%with a deterministic table $\encodedDB$ which is a set of $\numvar$ tuples, encoding the set of possible worlds $\idb$.  The probability distribution $\pd$ over the set of database instances (possible worlds) is the one
%being the distribution induced from the requirement that each tuple $\tup \in \encodedDB$ be treated as an independent Bernoulli distributed random variable with probability $\prob_\tup$.
%The possible worlds of a \abbrTIDB can be encoded by the vector $\vct{W}$, such that each of the $\numvar$ tuples in $\vct{W}$ has its own unique Bernoulli-distributed random variable, i.e. $\vct{W} = \inparen{W_{\tup_1},\ldots, W_{\tup_\numvar}}$, and for each tuple $\tup$, $\probOf(W_\tup) = \prob_\tup$.
%Given a vector $\vct{X}$ such that each $\tup \in \encodedDB$ has a unique formal variable annotation $X_\tup \in \vct{X}$, for a boolean domain $\{0,1\}^\numvar$, denote by $\pdb_{\vct{X}}$ the deterministic database consisting of exactly those tuples $\tup$ where $X_\tup = 1$.

%\BG{REMOVED:
%When $\pdb$ is a \abbrTIDB, for every output tuple $\tup$, $\query\inparen{\pdb}\inparen{\tup}$ can be encoded by a polynomial, with variables in $\vct{X}$.
%Green, Karvounarakis, and Tannen established (\cite{DBLP:conf/pods/GreenKT07}; see \Cref{fig:nxDBSemantics}) that for any $\raPlus$ query $\query$ and \abbrTIDB $\pdb$, there exists a polynomial $\poly_\tup\inparen{\vct{X}}$ following the standard addition and multiplication operators over Natural numbers (i.e., $\semN$-semiring semantics), such that $\query\inparen{\pdb_{\vct{W}}}\inparen{\tup} = \poly_\tup\inparen{\vct{W}}$.
%This in turn implies that $\expct\pbox{\query\inparen{\pdb}\inparen{\tup}} = \expct_{\vct{W}\sim\pd}\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$.}

Thanks to linearity of expectation, simple polynomial-time algorithms (for fixed query $\query$) exist for computing the expectation of a lineage polynomial $\apolyqdt$ when $\pdb$ is a \abbrTIDB and $\query$ is an $\raPlus$ query.
% The algo is trivial so I think putting in a 2010 cite seems like bit too much
%\cite{kennedy:2010:icde:pip})
% for computing exact results for bag-probabilistic count queries $Q$ over \abbrTIDB{}s.
However, it is also known that {\em deterministic} query processing for the same query $Q$ (over the deterministic database instance $\dbbase$) can also be done in polynomial time.
If our notion of efficiency were simply achieving a polynomial time algorithm, then we would be done.
However, in practice (and in theory), we care about the {\em fine-grained}/parameterized complexity of deterministic query processing (i.e. we care about the exact exponent in our polynomial runtime).


Given \abbrCTIDB $\pdb$ and query $\query$, let $\timeOf{}^*(Q,\pdb)$ denote the (optimal) runtime complexity of computing expected multiplicity (over all result tuples $\tup$). %\AR{Am changing these runtime definitions to include the runtime for all result tuples $\tup$.}
Denote by $\qruntime{Q, \db}$ the `runtime' of query $Q$ on deterministic database $\db$ under a cost model that is satisfied by a wide range of query processing algorithms, including those based on the recent work on worst-case optimal join algorithms (we make this runtime concrete in \Cref{sec:gen}). %\AR{We need to move the definition of $\qruntime{}$ to \Cref{sec:background} because among others we now need it in our lower bound arguments as well.}).
%Denoting by $\dbbase = \bigcup_{\db \in \idb} \db$ the set of all possible tuples in \abbrPDB $\pdb = \inparen{\idb, \pd}$,
\sout{
We finally have all the pieces to state a formal specification of our problem:
}
\AHchange{
Given the above, the natural question to ask is whether or not it is always the case that $\timeOf{}^*\inparen{\query, \pdb}\leq\qruntime{\query, \dbbase}$?
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{Problem}\label{prob:informal}
%Given an $\raPlus$ query $\query$ and \abbrTIDB
%% OK: added motivation
%%\AR{Changed this to \abbrTIDB: we should motivate why we are restricting ourselves to this special case here.}
%\abbrBPDB $\pdb$, is it the case that $\timeOf{}^*(Q,\pdb) \le  O(\qruntime{Q, \dbbase})$?
%\end{Problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% However the question remains: \emph{can bag-probabilistic databases be as fast as deterministic queries}.
%In this paper, we explore the \emph{fine-grained complexity} of bag-probabilistic database query evaluation.

%Atri: I'm not sure if this comment makes much sense here-- it sort of breaks the flow I think. I'll refer to this when talking about our results.
%The problem of deterministic query evaluation is known to be \sharpwonehard\footnote{A problem is in \sharpwone if the runtime of the most efficient known algorithm to solve it is lower bounded by some function $f$ of a parameter $k$, where the growth in runtime is polynomially dependent on $f(k)$, i.e. $\Omega\inparen{\numvar^{f(k)}}$.} in data complexity for general $\query$.  For example, the counting $k$-cliques query problem (where the parameter $k$ is the size of the clique) is \sharpwonehard since (under standard complexity assumptions) it cannot run in time faster than $n^{f(k)}$ for some strictly increasing $f(k)$.

%In this paper, we begin to explore whether the problem of bag-probabilistic query evaluation (which we relate to deterministic query processing more precisely below) falls into this same complexity class.
\AHchange{
This question
}
 is a special case of computing the expected multiplicity of $\tup$ since we are asking whether the query evaluation over a \abbrCTIDB is {\em linear} in the runtime of deterministic query processing.
We stress that this question is very well motivated, even for \abbrTIDBs: An answer \AHchange{to the above question} in the affirmative \AH{not sure that this is the best way of putting it} indicates that bag-probabilistic databases can be competitive with deterministic databases, opening the door for deployment in practice.






\mypar{Approximating the expected multiplicities}
%\AR{Have done my pass till here}
Our negative results indicate that \abbrBPDB{}s can not achieve comparable performance to deterministic databases for exact results (under complexity assumptions). In fact, under plausible hardness conjectures, one cannot (drastically) improve upon the trivial algorithm to exactly compute the expected multiplicities for \abbrTIDB. A natural followup is whether we can do better if we are willing to settle for an approximation to the expected multiplities.
In the remainder of this work, we demonstrate that a $(1\pm\epsilon)$ (multiplicative) approximation with competitive performance is achievable.

\input{two-step-model}\AR{In \Cref{fig:two-step}, perhaps in caption give f/w pointer to definition of $Q$?}

We adopt the two-step intensional model of query evaluation used in set-\abbrPDB\xplural, as illustrated in \Cref{fig:two-step}:
(i) \termStepOne (\abbrStepOne): Given input $\dbbase$ and $\query$, output every tuple $\tup$ that possibly satisfies $\query$, annotated with its lineage polynomial ($\poly(\vct{X})=\apolyqdt\inparen{\vct{X}}$);
(ii) \termStepTwo (\abbrStepTwo): Given $\poly(\vct{X})$ for each tuple, compute $\expct\pbox{\poly(\vct{\randWorld})}$.
Let $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit)$ denote the runtime of \abbrStepOne when it outputs $\circuit$ (which is a representation of $\poly$ as an arithmetic circuit --- more on this representation shortly).
Denote by $\timeOf{\abbrStepTwo}(\circuit)$ (recall $\circuit$ is the output of \abbrStepOne) the runtime of \abbrStepTwo, allowing us to formally define our objective:\AH{What if there are more efficient representations than circuits?}
\AHchange{
Our next question is whether or not there exists a $\inparen{1\pm\epsilon}$-approximation algorithm that is linear to the deterministic query?  If so, we have shown that approximation of bag \abbrPDB\xplural is comparable to deterministic query processing.
}

%\sout{
\begin{Problem}\label{prob:big-o-joint-steps}
Given \abbrBPDB $\pdb$, $\raPlus$ query $\query$,
is there a $(1\pm\epsilon)$-approximation of $\expct_{\db\sim\pd}\pbox{\query\inparen{\db}\inparen{\tup}}$ for all result tuples $\tup$ where
$\exists \circuit : \timeOf{\abbrStepOne}(Q,\dbbase,\circuit) + \timeOf{\abbrStepTwo}(\circuit) \le O_\epsilon(\qruntime{Q, \dbbase})$?
\end{Problem}
%Note that if the answer to the above problem is yes, then we have shown that the answer to \Cref{prob:informal} is yes (when we are interested in approximating the expected multiplicities).
%}

We show in \Cref{sec:circuit-depth} %{sec:gen}\AR{Refs needs to be updated}
%\OK{confirm this ref}
%Atri: fixed the ref
 an $O(\qruntime{Q, \dbbase})$ algorithm for constructing the lineage polynomial for all result tuples of an $\raPlus$ query $\query$ (or more more precisely, a single circuit $\circuit$ with one sink per tuple representing the tuple's lineage).
% , and by extension the first step is in \sharpwonehard\AH{\sharpwonehard is not defined.}.
A key insight of this paper is that the representation of $\circuit$ matters.
For example, if we insist that $\circuit$ represent the lineage polynomial in the standard monomial basis (henceforth, \abbrSMB)\footnote{
  This is the representation, typically used in set-\abbrPDB\xplural, where the polynomial is reresented as sum of `pure' products. See \Cref{def:smb} for a formal definition.
}, the answer to the above question in general is no, since then we will need $\abs{\circuit}\ge \Omega\inparen{\inparen{\qruntime{Q, \dbbase}}^k}$,
%\BG{should be $|\idb |$?},
%Atri: No, this is fine
and hence, just $\timeOf{\abbrStepOne}(Q,\dbbase,\circuit)$ will be too large.

However, systems can directly emit compact, factorized representations of $\poly(\vct{X})$ (e.g., as a consequence of the standard projection push-down optimization~\cite{DBLP:books/daglib/0020812}).
For example, in~\Cref{fig:two-step}, $B(Y+Z)$ is a factorized representation of the SMB-form $BY+BZ$.
Accordingly, this work uses (arithmetic) circuits\footnote{
  An arithmetic circuit is a DAG with variable and/or numeric source nodes and internal, each nodes representing either an addition or multiplication operator.
}
as the representation system of $\poly(\vct{X})$.


% When $\poly(\vct{X})$ is in standard monomial basis (\abbrSMB)\footnote{A polynomial is in \abbrSMB when it is a sum of products of variables (a variable can occur more than once), where each product of variables is unique.}, by linearity of expectation and independence of \abbrTIDB, it follows that $\timeOf{\abbrStepTwo}(Q,\pdb)$ is $O(|\poly_\tup(\vct{X})|)$ and thus also $\bigO{\timeOf{\abbrStepOne}(Q,\pdb)}$.
% \AH{Is this obvious enough for the typical reviewer to realize?}
%   Recall that $\prob_i$ denotes the probability of tuple $\tup_i$ (i.e. $\probOf\pbox{W_i = 1}$) for $i \in [\numvar]$.  Consider another special case when for all $i$ in $[\numvar]$, $\prob_i = 1$.
% % Replaced the stuff below with something more auccint
% %For output tuple $\tup'$ of $\query\inparen{\pdb}$, computing $\expct\pbox{\poly_{\tup'}\inparen{\vct{\randWorld}}}$ is linear in
% %$\abs{\poly_\tup}$
% %the size of the arithemetic circuit
% %, since we can essentially push expectation through multiplication of variables dependent on one another.\footnote{For example in this special case, computing $\expct\pbox{(X_iX_j + X_\ell X_k)^2}$ does not require product expansion, since we have that $p_i^h x_i^h = p_i \cdot 1^{h-1}x_i^h$.}
% In this case, we have for any output tuple $\tup$, $\expct\pbox{\poly(\vct{W})}=\Phi(1,\dots,1)$.
% Thus, we have another case where $\timeOf{\abbrStepTwo}(Q,\pdb)$ is $\bigO{\timeOf{\abbrStepOne}(Q,\pdb)}$ and we again achieve deterministic query runtime for $\query\inparen{\pdb}$ (up to a constant factor).  These observations introduce our first formalization of~\Cref{prob:informal}:


Given that there exists a representation $\circuit^*$ such that $\timeOf{\abbrStepOne}(\query,\dbbase,\circuit^*)\le O(\qruntime{\query, \dbbase})$, we can now focus on the complexity of \abbrStepTwo.
We can represent the factorized lineage polynomial by its correspoding arithmetic circuit $\circuit$ (whose size we denote by $|\circuit|$).
%\BG{This sentence didn't parse for me. What do we mean by representing a polynomial by a size?}
%Atri: fixed
As we also show in \Cref{sec:circuit-runtime}, this size is also bounded by $\qruntime{Q, \dbbase}$ (i.e., $|\circuit^*| \le O(\qruntime{Q, \dbbase})$).
Thus, \AHchange{the question of approximation} %\Cref{prob:big-o-joint-steps} 
can be reframed as:

%Atri: Replaced the text below by the above. I know I had talked about $|\circuit|^k$ but I think the stuff below breaks the flow a bit
%Re-stating our earlier observation, given a circuit \circuit, if \circuit is in \abbrSMB (i.e. every sink to source path has a prefix of addition nodes and the rest of the internal nodes are multiplication nodes), then we have that $\timeOf{\abbrStepTwo}(Q,\pdb)$ is indeed $\bigO{\timeOf{\abbrStepOne}(Q,\pdb)}$.  We note that \abbrSMB representations are produced by queries with a projection operation on top of a join operation.
% the form $\project, \project\inparen{\join},$ etc.
% Suppose, on the contrary, that \circuit is not in \abbrSMB and rather in some factorized form.  Then to naively compute \abbrStepTwo, one needs to convert \circuit into \circuit' such that \circuit' is in \abbrSMB, and then compute $\expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$, which takes $\bigO{|\circuit|^k}$ time for the case that $k$ is the degree of the polynimial $\Phi_\tup(\vct{X})$.  Since $|\circuit'|$ lies between $\bigO{|\circuit|}$ and $\bigO{|\circuit|^k}$, it behooves us to determine which of these extremes is true for the general \circuit.  This leads us to the main problem statement of our paper:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Problem}\label{prob:intro-stmt}
Given one circuit $\circuit$ that encodes $\apolyqdt$ for all result tuples $\tup$ (one sink per $\tup$) for \abbrBPDB $\pdb$ and $\raPlus$ query $\query$, does there exist an algorithm that computes a $(1\pm\epsilon)$-approximation of $\expct_{\db\sim\pd}\pbox{\query\inparen{\db}\inparen{\tup}}$ (for all result tuples $\tup$) in $\bigO{|\circuit|}$ time?
%\OK{This doesn't parse.  What is $\bigO{\abbrStepOne}$?  Should this be $\bigO{\poly}$?}
\end{Problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%
%Contributions, Overview, Paper Organization
%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\mypar{Overview of our Techniques} All of our results rely on working with a {\em reduced} form of the lineage polynomial $\poly$. In fact, it turns out that for the \abbrTIDB (and \abbrBIDB) case, computing the expected multiplicity is {\em exactly} the same as evaluating this reduced polynomial over the probabilities that define the \abbrTIDB/\abbrBIDB. Next, we motivate this reduced polynomial.
Consider the query $\query$ defined as follows over the bag relations of \Cref{fig:two-step}:
\begin{lstlisting}
SELECT 1 FROM OnTime a, Route r, OnTime b
WHERE a.city = r.city1 AND b.city = r.city2
\end{lstlisting}
%$Q()\dlImp$$OnTime(\text{City}), Route(\text{City}, \text{City}'),$ $OnTime(\text{City}')$
It can be verified that $\poly\inparen{A, B, C, E, X, Y, Z}$ for the sole result tuple (i.e. the count) of $\query$ is $AXB + BYE + BZC$. Now consider the product query $\query^2 = \query \times \query$.
%\BG{$\query(\db)$ is a query result, so I changed it to this}
%Atri: Sounds good.

The lineage polynomial for $Q^2$ is given by $\poly^2\inparen{A, B, C, E, X, Y, Z}$:%\AR{Changed the variable $D$ to $E$ to avoid conflict with use of $D$ as a DB.}
$$%\begin{multline*}
%\inparen{AXB + BYE + BZC}^2\\
=A^2X^2B^2 + B^2Y^2E^2 + B^2Z^2C^2 + 2AXB^2YE + 2AXB^2ZC + 2B^2YEZC.
$$%\end{multline*}
By exploiting linearity of expectation, further pushing expectation through independent \abbrTIDB variables and observing that for any $\randWorld\in\{0, 1\}$, we have $\randWorld^2=\randWorld$, the expectation is
%\AH{If we choose to use $\pd$ in \Cref{prob:bag-pdb-poly-expected}, then we either need to follow the same convention here OR introduce the notation $\pdassign$ before using it.}
$\expct\limits_{\vct{\randWorld}\sim\pdassign}\pbox{\poly^2\inparen{\vct{\randWorld}}}$ (where $\randWorld_A$ is the random variable corresponding to $A$, distributed as $\pdassign$).

%Atri: Combined the the first step below with the next one to save space.
%\begin{footnotesize}
%\begin{multline*}
%\expct\pbox{\randWorld_A^2}\expct\pbox{\randWorld_X^2}\expct\pbox{\randWorld_B^2} + \expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Y^2}\expct\pbox{\randWorld_E^2} + \expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Z^2}\expct\pbox{\randWorld_C^2} + 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_X}\expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E}\\
%+ 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C} + 2\expct\pbox{\randWorld_B^2}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C}.
%\end{multline*}
%\end{footnotesize}
%\noindent Since for any $\randWorld\in\{0, 1\}$, we have $\randWorld^2=\randWorld$,
%then for any $k > 0$, $\expct\pbox{\randWorld^k} = \expct\pbox{\randWorld}$, which means that
%$\expct\limits_{\vct{\randWorld}\sim\pdassign}\pbox{\poly^2\inparen{\vct{\randWorld}}}$ simplifies to:
\begin{footnotesize}
\begin{multline*}
\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_X}\expct\pbox{\randWorld_B} + \expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E} + \expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C} + 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_X}\expct\pbox{\randWorld_B}\expct{\randWorld_Y}\expct\pbox{\randWorld_E} \\
+ 2\expct\pbox{\randWorld_A}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C} + 2\expct\pbox{\randWorld_B}\expct\pbox{\randWorld_Y}\expct\pbox{\randWorld_E}\expct\pbox{\randWorld_Z}\expct\pbox{\randWorld_C}.
\end{multline*}
\end{footnotesize}
\noindent This property leads us to consider a structure related to the lineage polynomial.
\begin{Definition}\label{def:reduced-poly}
For any polynomial $\poly(\vct{X})$ corresponding to a \abbrTIDB (henceforth, \abbrTIDB-lineage polynomial),
%\BG{Better introduce the notion of TIDB lin poly before here, then it iis more clear?},
%Atri: Done
 define the \emph{reduced polynomial} $\rpoly(\vct{X})$ to be the polynomial obtained by setting all exponents $e > 1$ in the \abbrSMB form of $\poly(\vct{X})$ to $1$.
\end{Definition}
With $\poly^2\inparen{A, B, C, E, X, Y, Z}$ as an example, we have:
\begin{align*}
&\widetilde{\poly^2}(A, B, C, E, X, Y, Z) = AXB + BYE + BZC + 2AXBYE + 2AXBZC + 2BYEZC.
%&\; = AXB + BYD + BZC + 2AXBYD + 2AXBZC + 2BYDZC
\end{align*}
Note that we have argued that for our specific example the expectation that we want is $\widetilde{\poly^2}(\probOf\inparen{A=1},$ $\probOf\inparen{B=1}, \probOf\inparen{C=1}), \probOf\inparen{E=1}, \probOf\inparen{X=1}, \probOf\inparen{Y=1}, \probOf\inparen{Z=1})$.
%It can be verified that the reduced polynomial parameterized with each variable's respective marginal probability is a closed form of the expected count (i.e., $\expct\limits_{\vct{\randWorld}\sim\pd}\pbox{\Phi^2\inparen{\vct{X}}} = \widetilde{\Phi^2}(\probOf\pbox{A=1},$ $\probOf\pbox{B=1}, \probOf\pbox{C=1}), \probOf\pbox{D=1}, \probOf\pbox{X=1}, \probOf\pbox{Y=1}, \probOf\pbox{Z=1})$).
\Cref{lem:tidb-reduce-poly} generalizes the equivalence to {\em all} $\raPlus$ queries on \abbrTIDB\xplural (proof in \Cref{subsec:proof-exp-poly-rpoly}).
\begin{Lemma}\label{lem:tidb-reduce-poly}
Let $\pdb$ be a \abbrTIDB over $n$ input tuples
 such that the probability distribution $\pdassign$ over $\vct{W}\in\{0,1\}^\numvar$ (the set of possible worlds) is induced by the probability vector $\probAllTup = \inparen{\prob_1,\ldots,\prob_\numvar}$ where $\prob_i=\probOf\inparen{W_i=1}$.
For any \abbrTIDB-lineage polynomial
%\BG{Term has not been introduced yet.}
%Atri: fixed
 $\poly\inparen{\vct{X}}=\apolyqdt(\vct{X})$, it holds that $
	\expct_{\vct{W} \sim \pdassign}\pbox{\poly\inparen{\vct{W}}} = \rpoly\inparen{\probAllTup}.
$
\end{Lemma}

To prove our hardness result we show that for the same $Q$ from the example above, for an arbitrary `product width' $k$, the query $Q^k$ is able to encode various hard graph-counting problems (assuming $\bigO{\numvar}$ tuples rather than the $O(1)$ tuples in \Cref{fig:two-step}).
We do so by considering an arbitrary graph $G$ (analogous to the $Route$ relation of $\query$) and analyzing how the coefficients in the (univariate) polynomial $\widetilde{\poly}\left(p,\dots,p\right)$ relate to counts of subgraphs in $G$ that are isomorphic to various graphs with $k$ edges. E.g., we exploit the fact that the leading coefficient in $\poly$ corresponding to $\query^k$ is proportional to the number of $k$-matchings in $G$, a known hard problem in parameterized/fine-grained complexity literature.

For an upper bound on approximating the expected count, it is easy to check that if all the probabilties are constant then $\poly\left(\prob_1,\dots, \prob_n\right)$ (i.e. evaluating the original lineage polynomial over the probability values) is a constant factor approximation.  For example, using $\query^2$ from above, using $\prob_A$ to denote $\probOf\pbox{A = 1}$ (and similarly for the other variables), we can see that
\begin{footnotesize}
\begin{align*}
\hspace*{-3mm}
	\poly^2\inparen{\probAllTup} &= \prob_A^2\prob_X^2\prob_B^2 + \prob_B^2\prob_Y^2\prob_E^2 + \prob_B^2\prob_Z^2\prob_C^2 + 2\prob_A\prob_X\prob_B^2\prob_Y\prob_E + 2\prob_A\prob_X\prob_B^2\prob_Z\prob_C + 2\prob_B^2\prob_Y\prob_E\prob_Z\prob_C\\
	&\leq\prob_A\prob_X\prob_B + \prob_B\prob_Y\prob_E + \prob_B\prob_Z\prob_C +
2\prob_A\prob_X\prob_B\prob_Y\prob_E + 2\prob_A\prob_X\prob_B\prob_Z\prob_C + 2\prob_B\prob_Y\prob_E\prob_Z\prob_C
	= \rpoly\inparen{\vct{p}}
	 %\inparen{0.9\cdot 1.0\cdot 1.0 + 0.5\cdot 1.0\cdot 1.0 + 0.5\cdot 1.0\cdot 0.5}^2 = 2.7225 < 3.45 = \rpoly^2\inparen{\probAllTup}
\end{align*}
\end{footnotesize}
If we assume that all seven probability values are at least $p_0>0$,
%Choose the least factor that is reduced in $\rpoly^2\inparen{\vct{X}}$, in this case $\prob_A\prob_X\prob_B$, and
we get that $\poly^2\inparen{\vct{\prob}}$ is in the range $[\inparen{p_0}^3\cdot\rpoly\inparen{\vct{\prob}}, \rpoly\inparen{\vct{\prob}}]$.
%
To get an $(1\pm \epsilon)$-multiplicative approximation we uniformly sample monomials from the \abbrSMB representation of $\poly$ and `adjust' their contribution to $\widetilde{\poly}\left(\cdot\right)$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\mypar{Applications}
Recent work in heuristic data cleaning~\cite{yang:2015:pvldb:lenses,DBLP:journals/vldb/SaRR0W0Z17,DBLP:journals/pvldb/RekatsinasCIR17,DBLP:journals/pvldb/BeskalesIG10,DBLP:journals/vldb/SaRR0W0Z17} emits a \abbrPDB when insufficient data exists to select the `correct' data repair.
Probabilistic data cleaning is a crucial innovation, as the alternative is to arbitrarily select one repair and `hope' that queries receive meaningful results.
Although \abbrPDB queries instead convey the trustworthiness of results~\cite{kumari:2016:qdb:communicating}, they are impractically slow~\cite{feng:2019:sigmod:uncertainty,feng:2021:sigmod:efficient}, even in approximation (see \Cref{sec:karp-luby}).  
Bags, as we consider, are sufficient for production use, where bag-relational algebra is already the default for performance reasons.
Our results show that bag-\abbrPDB\xplural can be competitive, laying the groundwork for probabilistic functionality in production database engines.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mypar{Paper Organization} We present relevant background and notation in \Cref{sec:background}. We then prove our main hardness results in \Cref{sec:hard} and present our approximation algorithm in \Cref{sec:algo}. %We present some (easy) generalizations of our results in \Cref{sec:gen}.
%and also discuss extensions from computing expectations of polynomials to the expected result multiplicity problem
%\AH{I don't think I understand what the sentence (about extensions) is saying.}
% (\Cref{def:the-expected-multipl}).
Finally, we discuss related work in \Cref{sec:related-work} and conclude in \Cref{sec:concl-future-work}.  All proofs are in the appendix. 

%No reviewer comments in arxiv submission.
%Our responses to ICDT first cycle reviewer comments are in \Cref{sec:rebuttal}. % the appendix.\AR{Would be good to have a specific app ref to rebuttal}


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