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\section{Outline of 1st ICDT Introduction Submission}

\begin{outline}[enumerate]
	\1 Problem Introduction and Background
		\2 Set-\abbrPDB notation, concepts, common (known) results
			\3 Dichotomy
			\3 Exact Computation \sharpphard
		\2 Formal definition of expected result multiplicity
		\2 Example
			\3 Assumed setting of {\emph set inputs}
			\3 Example based on explaining and motivating formal definition of expected result multiplicity
	\1 Discussion of set-\abbrPDB\xplural
		\2 Lineage from PosBool$[\vct{X}]$
		\2 Encoding of possible worlds via $\vct{X}$
		\2 Computing probability can be done using only the lineage
	\1 Discussion of bag-\abbrPDB\xplural 
		\2 Link to $\domN[\vct{X}]$
		\2 Link to computing the expected count of a lineage polynomial
		\2 Example to illustrate computing an expected count over a lineage polynomial
	\1 Computing expected multiplicity for an \abbrSOP representation versus a factorized representation
		\2 Linear for \abbrSOP
		\2 Introduce the problem by asking if it's linear in the size of the representation for factorized representation produced by such query optimizations as projection push-down.
		\2 State our theoretical results (informally) that it is not linear in general
	\1 Contributions
		\2 Hardness results for the expected multiplicity problem
			\3 Reduction to counting $\kElem$-matchings
		\2 Introduce our approximation algorithm and its guarantees
		\2 Generalization to bag-\abbrPDB\xplural
		\2 Result over $\raPlus$ queries
		\2 Higher moments
	\1 Overview of our techniques
		\2 Informal introduction to $\rpoly$ with example
		\2 Definition of reduced polynomial
		\2 Equivalence of $\rpoly$ and computing $\expct$
		\2 Further details into the technique of obtaining our hardness result
	\1 Paper organization
		\2 Also includes evaluation semantics figure
\end{outline}