84 lines
7.2 KiB
TeX
84 lines
7.2 KiB
TeX
%root:main.tex
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%!TEX root=./main.tex
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\section{Hardness of exact computation}
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\label{sec:hard}
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In this section, we will prove that computing $\expct\limits_{\vct{W} \sim \pd}\pbox{\poly(\vct{W})}$ 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 $\semNX$-PDBs under bag semantics. 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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Preliminaries}
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Our hardness results are based on (exactly) counting the number of occurrences of a subgraph $H$ in $G$. Let $\numocc{G}{H}$ denote the number of occurrences of $H$ in graph $G$. 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.
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In particular, we will consider the problems of computing the following counts (given $G$ as an input and 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). Our hardness result in \Cref{sec:multiple-p} is based on the following result:
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\begin{Theorem}[\cite{k-match}]
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\label{thm:k-match-hard}
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Given positive integer $k$ and undirected graph $G$ with no self-loops or parallel edges, computing $\numocc{G}{\kmatch}$ exactly is %counting the number of $k$-matchings in $G$ is
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\sharpwonehard (parameterization is in $k$).
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\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 $k$. In fact, all known algorithms to solve this problem take time $|\vset|^{\Omega(k)}$.
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%
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Our hardness result in Section~\ref{sec:single-p} is based on the following conjectured hardness result:
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\begin{hypo}
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\label{conj:graph}
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There exists a constant $\eps_0>0$ such that given an undirected graph $G=(\vset,\edgeSet)$, computing exactly $\numocc{G}{\tri}$ cannot be done in time $o\inparen{|\edgeSet|^{1+\eps_0}}$.
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\end{hypo}
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%
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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 $\eps_0\ge \frac 13$.
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%The current best known algorithm to count the number of $3$-matchings, to
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%\AR{Need to add something about 3-paths and 3-matchings as well.}
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Both of our hardness results rely on a simple query polynomial encoding of the edges of a graph.
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To prove our hardness result, consider a graph $G(\vset, \edgeSet)$, where $|\edgeSet| = m$, $|\vset| = \numvar$. Our query polynomial has a variable $X_i$ for every $i$ in $[\numvar]$.
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Consider the polynomial
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$\poly_{G}(\vct{X}) = \sum\limits_{(i, j) \in \edgeSet} X_i \cdot X_j.$
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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}
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For any graph $G=([n],\edgeSet)$ and $\kElem\ge 1$, define
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\[\poly_{G}^\kElem(X_1,\dots,X_n) = \left(\sum\limits_{(i, j) \in \edgeSet} X_i \cdot X_j\right)^\kElem\]
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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
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even hold for the expression trees. %this polynomial can be encoded in an expression tree of size $\Theta(km)$.
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\noindent Returning to \Cref{fig:ex-shipping-simp}, it is easy to see that $\poly_{G}^\kElem(\vct{X})$ generalizes our running example query:
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\resizebox{1\linewidth}{!}{
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\begin{minipage}{1.05\linewidth}
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\[\poly^k_G\dlImp OnTime(C_1),Route(C_1, C_1'),OnTime(C_1'),\dots,OnTime(C_\kElem),Route(C_\kElem,C_\kElem'),OnTime(C_\kElem')\]
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\end{minipage}
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}
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where adapting the PDB instance in \Cref{fig:ex-shipping-simp}, relation $OnTime$ has $n$ tuples corresponding to each vertex in $\vset=[n]$ each with probability $\prob$ and $Route(\text{City}_1, \text{City}_2)$ 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.}
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Note that this implies that our hard query polynomial can be represented as an expression tree produced by a project-join query with same probability value for each input tuple $\prob_i$.
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\subsection{Multiple Distinct $\prob$ Values}
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\label{sec:multiple-p}
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%Unless otherwise noted, all proofs for this section are in \Cref{app:single-mult-p}.
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We are now ready to present our main hardness result.
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Theorem}\label{thm:mult-p-hard-result}
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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 \sharpwonehard (parameterization is in $k$).
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\end{Theorem}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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We will prove the above result by reducing from the problem of computing the number of $k$-matchings in $G$. Given the current best-known algorithm for this counting problem, our results imply that unless the state-of-the-art $k$-matching algorithms are improved, we cannot hope to solve our problem in time better than $\Omega_k\inparen{m^{k/2}}$ where $m=\abs{\edgeSet}$, which is only quadratically faster than expanding $\poly_{G}^\kElem(\vct{X})$ into its \abbrSMB form and then using \Cref{cor:expct-sop}. The approximation algorithm we present in \Cref{sec:algo} has runtime $O_k\inparen{m}$ for this query. % (since it runs in linear-time on all lineage polynomials).
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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}):
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\begin{Lemma}\label{lem:qEk-multi-p}
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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.
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\end{Lemma}
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