paper-BagRelationalPDBsAreHard/mult_distinct_p.tex

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\section{Hardness of exact computation}
\label{sec:hard}

In this section, we will prove 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:intro-stmt} 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. 

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\subsection{Preliminaries}
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 runtime of computing $\numocc{G}{\kmatch}$.  Our hardness result in \Cref{sec:multiple-p} is based on the following result:

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\begin{Theorem}[\cite{k-match}]
\label{thm:k-match-hard}
Given positive integer $k$ and undirected graph $G$ with no self-loops or parallel edges, the time $\kmatchtime$ to compute $\numocc{G}{\kmatch}$ exactly is $\littleomega{f(k)\cdot \numedge^c}$ for any function $f$ and fixed constant $c$ independent of $\numedge$ and $k$. %counting the number of $k$-matchings in $G$ is\sharpwonehard (parameterization is in $k$).
\end{Theorem}
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%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 $k$. 
\begin{hypo}[\cite{k-match}]\label{conj:known-algo-kmatch}
All best known algorithms to solve $\numocc{G}{\kmatch}$ take time $\kmatchtime = \bigO{|\vset|^{\Omega(k)}}$.
\end{hypo}
%
Our hardness result in Section~\ref{sec:single-p} is based on the following conjectured hardness result:
%
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\begin{hypo}
\label{conj:graph}
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 $o\inparen{|\edgeSet|^{1+\eps_0}}$.
\end{hypo}
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%
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 $\eps_0\ge \frac 13$.
%The current best known algorithm to count the number of $3$-matchings, to
%\AR{Need to add something about 3-paths and 3-matchings as well.}

Both 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 $[\numvar]$.
Consider the polynomial
$\poly_{G}(\vct{X}) = \sum\limits_{(i, j) \in \edgeSet} X_i \cdot X_j.$
The hard polynomial for our problem will be a suitable power $k\ge 3$ of the polynomial above:
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\begin{Definition}\label{def:qk}
For any graph $G=(V,\edgeSet)$ and $\kElem\ge 1$, define
\[\poly_{G}^\kElem(X_1,\dots,X_n) = \left(\sum\limits_{(i, j) \in \edgeSet} X_i \cdot X_j\right)^\kElem\]
\end{Definition}
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%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 $\poly_{G}^\kElem(\vct{X})$ generalizes our example query from the introduction. Let us alias 
\begin{lstlisting}
SELECT 1 FROM OnTime a, Route r, OnTime b
WHERE a.city = r.city1 AND b.city = r.city2
\end{lstlisting}
as $R_i$ for each $i \in [k]$.  The query $\query^k$ then becomes
\begin{lstlisting}
SELECT 1 FROM $R_1$ JOIN $R_2$ JOIN$\cdots$JOIN $R_k$
\end{lstlisting}          
%RA format for the same query
%\begin{align*}
%\query^k_G \coloneqq &\inparen{\project_\emptyset\inparen{OnTime \join_{City = City_1} Route \join_{{City}_2 = City'}\rename_{City' \leftarrow City}(OnTime)}}\times_2\cdots\\
%&\cdots \times_k \inparen{\project_\emptyset\inparen{OnTime \join_{City = City_1} Route \join_{{City}_2 = City'}\rename_{City' \leftarrow City}(OnTime)}}
%\end{align*}

%\resizebox{1\linewidth}{!}{
%\begin{minipage}{1.05\linewidth}
%\[\poly^k_G\dlImp OnTime(C_1),Route(C_1, C_1'),OnTime(C_1'),\dots,OnTime(C_\kElem),Route(C_\kElem,C_\kElem'),OnTime(C_\kElem')\]
%\end{minipage}
%}
\noindent where adapting the PDB instance in \Cref{fig:two-step}, relation $OnTime$ has $4$ tuples corresponding to each vertex for $i$ in $[4]$, 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.}
Note that this implies that 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.

\begin{Lemma}\label{lem:tdet-om}
For the $\query^k$ of \Cref{def:qk}, the runtime $\qruntime{\query^k, \dbbase}$ is $O_k(\numedge)$.
\end{Lemma}

%\begin{Corollary}\label{cor:at-least-kmatch}
%\end{Corollary}
%\begin{proof}[Proof of \Cref{cor:at-least-kmatch}
%\end{proof}
%
%\begin{Corollary}\label{cor:best-curr-algo}
%\end{Corollary}
%\begin{proof}[Proof of \Cref{cor:best-curr-algo}
%\end{proof}

\subsection{Multiple Distinct $\prob$ Values}
\label{sec:multiple-p}
%Unless otherwise noted, all proofs for this section are in \Cref{app:single-mult-p}.
We are now ready to present our main hardness result.
%

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\begin{Theorem}\label{thm:mult-p-hard-result}
Let $\prob_0,\ldots,\prob_{2k}$ be $2k + 1$ distinct values in $(0, 1]$.  Then computing $\rpoly_G^\kElem(\prob_i,\dots,\prob_i)$ for arbitrary $G$
%and any $(2k+1)$ distinct values $\prob_i$ ($0\le i \le 2k$)
is in time $\bigOmega{\kmatchtime}$.
\end{Theorem}
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%
The second row of \Cref{tab:lbs} is upheld by \Cref{thm:mult-p-hard-result}, \Cref{lem:tdet-om}, and \Cref{thm:k-match-hard}.  The third row is proved by \Cref{thm:mult-p-hard-result}, \Cref{lem:tdet-om}, and \Cref{conj:known-algo-kmatch}.
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%NEEDS to be moved to appendix
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%\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.
%\end{Lemma}
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%END move to appendix
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