In this section, we will prove the hardness results claimed in Table~\ref{tab:lbs} for a specific (family) of hard instance $(\query,\pdb)$ for \Cref{prob:bag-pdb-poly-expected} where $\pdb$ is a \abbrTIDB.
% that computing $\expct\pbox{\poly(\vct{W})}$ exactly for a \ti-lineage polynomial $\poly(\vct{X})$ generated from a project-join query (even an expression tree representation) is \sharpwonehard.
Note that this implies hardness for \bis and general \abbrBPDB, answering \Cref{prob:bag-pdb-poly-expected} (and hence the equivalent \Cref{prob:bag-pdb-query-eval}) in the negative.
%Furthermore, we demonstrate in \Cref{sec:single-p} that the problem remains hard, even if $\probOf[X_i=1] = \prob$ for all $X_i$ and any fixed valued $\prob \in (0, 1)$ as long as certain popular hardness conjectures in fine-grained complexity hold.
Our hardness results are based on (exactly) counting the number of (not necessarily induced) subgraphs in $G$ isomorphic to $H$. Let $\numocc{G}{H}$ denote this quantity. We can think of $H$ as being of constant size and $G$ as growing. %In query processing, $H$ can be viewed as the query while $G$ as the database instance.
In particular, we will consider the problems of computing the following counts (given $G$ in its adjacency list representation): $\numocc{G}{\tri}$ (the number of triangles), $\numocc{G}{\threedis}$ (the number of $3$-matchings), and the latter's generalization $\numocc{G}{\kmatch}$ (the number of $k$-matchings). We use $\kmatchtime$ to denote the optimal runtime of computing $\numocc{G}{\kmatch}$. Our hardness results in \Cref{sec:multiple-p} is based on the following hardness results/conjectures:
Given positive integer $k$ and undirected graph $G=(\vset,\edgeSet)$ with no self-loops or parallel edges, the time $\kmatchtime$ to compute $\numocc{G}{\kmatch}$ exactly is $\littleomega{f(k)\cdot |\edgeSet|^c}$ for any function $f$ and fixed constant $c$ independent of $\numedge$ and $k$ (assuming $\sharpwzero\ne\sharpwone$. %counting the number of $k$-matchings in $G$ is\sharpwonehard (parameterization is in $k$).
%The above result means that we cannot hope to count the number of $k$-matchings in $G=(\vset,\edgeSet)$ in time $f(k)\cdot |\vset|^{c}$ for any function $f$ and constant $c$ independent of $k$.
We note that the above conjecture is somewhat non-standard. In particular, the best known state of the art algorithm to compute $\numocc{G}{\kmatch}$ takes time $\Omega\inparen{|V|^{k/2}}$ (i.e. if this is the best algorithm then $c_0=\frac14$)~\cite{k-match}. What the above conjecture is saying is that one can only hope for a polynomial improvement over the state of the art algorithm to compute $\numocc{G}{\kmatch}$.
There exists a constant $\eps_0>0$ such that given an undirected graph $G=(\vset,\edgeSet)$, computing $\numocc{G}{\tri}$ exactly cannot be done in time $o\inparen{|\edgeSet|^{1+\eps_0}}$.
Based on the so called {\em Triangle detection hypothesis} (cf.~\cite{triang-hard}), which states that detection of whether $G$ has a triangle or not takes time $\Omega\inparen{|\edgeSet|^{4/3}}$, implies that in Conjecture~\ref{conj:graph} we can take $\eps_0\ge\frac13$.
All of our hardness results rely on a simple lineage polynomial encoding of the edges of a graph.
To prove our hardness result, consider a graph $G=(\vset, \edgeSet)$, where $|\edgeSet| = m$, $\vset=[\numvar]$. Our lineage polynomial has a variable $X_i$ for every $i$ in $[\numvar]$.
%Our hardness results only need a \ti instance; We also consider the special case when all the tuple probabilities (probabilities assigned to $X_i$ by $\probAllTup$) are the same value. Note that our hardness results % do not require the general circuit representation and
%even hold for the expression trees. %this polynomial can be encoded in an expression tree of size $\Theta(km)$.
\noindent Returning to \Cref{fig:two-step}, it is easy to see that $\poly_{G}^\kElem(\vct{X})$ is the lineage polynomial corresponding to the query that generalizes our example query from \Cref{sec:intro}. Let us alias
\noindent Further, the PDB instance generalizes the one in \Cref{fig:two-step} as follows. Relation $OnTime$ has $n$ tuples corresponding to each vertex for $i$ in $[n]$, each with probability $\prob_i$ and $Route$ has tuples corresponding to the edges $\edgeSet$ (each with probability of $1$).\footnote{Technically, $\poly_{G}^\kElem(\vct{X})$ should have variables corresponding to tuples in $Route$ as well, but since they always are present with probability $1$, we drop those. Our argument also works when all the tuples in $Route$ also are present with probability $\prob$ but to simplify notation we assign probability $1$ to edges.}
In other words, for this instance $\dbbase$ contains the set of $n$ unary tuples in $OnTime$ (which corresponds to $\vset$) and $m$ binary tuples in $Route$ (which corresponds to $\edgeSet$).
Note that this implies that $\poly_{G}^\kElem$
%our hard lineage polynomial can be represented as an expression tree produced by a project-join query with same probability value for each input tuple $\prob_i$, and hence
is indeed a lineage polynomial for a \abbrTIDB\abbrPDB.
Let $\prob_0,\ldots,\prob_{2k}$ be $2k +1$ distinct values in $(0, 1]$. Then computing $\rpoly_G^\kElem(\prob_i,\dots,\prob_i)$ (over all $i\in[2k+1]$ for arbitrary $G=(\vset,\edgeSet)$
Note that the second row of \Cref{tab:lbs} follows from \Cref{prop:expection-of-polynom}, \Cref{thm:mult-p-hard-result}, \Cref{lem:tdet-om}, and \Cref{thm:k-match-hard} while the third row is proved by \Cref{prop:expection-of-polynom}, \Cref{thm:mult-p-hard-result}, \Cref{lem:tdet-om}, and \Cref{conj:known-algo-kmatch}. Since \Cref{conj:known-algo-kmatch} is non-standard, the latter hardness result should be interpreted as follows. Any substantial polynomial improvement for \Cref{prob:bag-pdb-poly-expected} (over the trivial algorithm that converts $\poly$ into SMB and then runs the obvious algorithm for \abbrStepTwo) would lead to an improvement over the state of the art {\em upper} bounds on $\kmatchtime$. Finally, note that \Cref{thm:mult-p-hard-result} needs one to be able to compute the expected multiplicities over $(2k+1)$ distinct values of $p_i$, each of which corresponds to distinct $\pd$ (for the same $\dbbase$), which explain the `Multiple' entry in the second column in the second and third row in \Cref{tab:lbs}. Next, we argue how to get rid of this latter requirement.
%\noindent The following lemma reduces the problem of counting $\kElem$-matchings in a graph to our problem (and proves \Cref{thm:mult-p-hard-result}):
%\begin{Lemma}\label{lem:qEk-multi-p}
%Let $\prob_0,\ldots, \prob_{2\kElem}$ be distinct values in $(0, 1]$. Then given the values $\rpoly_{G}^\kElem(\prob_i,\ldots, \prob_i)$ for $0\leq i\leq 2\kElem$, the number of $\kElem$-matchings in $G$ can be computed in $\bigO{\kElem^3}$ time.