finally finished the lb table
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@ -149,18 +149,24 @@ Given an $\raPlus$ query $\query$ and \abbrTIDB\AR{Changed this to \abbrTIDB: we
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We note that the above is a special case of \Cref{prob:bag-pdb-query-eval} since we are asking whether the query evaluation over \abbrBPDB is {\em linear} in the runtime of deterministic query processing time. We stress that this question is very well motivated. In particular,
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We note that the above is a special case of \Cref{prob:bag-pdb-query-eval} since we are asking whether the query evaluation over \abbrBPDB is {\em linear} in the runtime of deterministic query processing time. We stress that this question is very well motivated. In particular,
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we note that an answer in the affirmative for~\Cref{prob:informal} indicates that bag-probabilistic databases can be competitive with classical deterministic databases, opening the door for deployment in practice.
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we note that an answer in the affirmative for~\Cref{prob:informal} indicates that bag-probabilistic databases can be competitive with classical deterministic databases, opening the door for deployment in practice.
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Unfortunately, we prove the negative. In fact in \Cref{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 \Cref{prob:informal}.
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Unfortunately, we prove the negative. In fact in Table~\ref{tab:lbs}\AR{Cref was not formatting Table correct so added Table in explicitly.} we show that depending on what hardness result/conjecture we assume, we get various emphatic versions of {\em no} as an answer to \Cref{prob:informal}.
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\begin{table}
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\begin{table}
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\begin{tabular}{|c|c|c|}
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\begin{tabular}{|p{0.4\textwidth}|p{0.1\textwidth}|p{0.5\textwidth}|}
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\hline
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\hline
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Lower bound on $\timeOf{}^*(Q,\pdb)$ & How many $\pdb$s & Hardness Assumption\\
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Lower bound on $\timeOf{}^*(Q,\pdb)$ & Num. $\pd$s & Hardness Assumption\\
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\hline
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\hline
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$(\qruntime{Q, \dbbase})^{1+\eps_0}$ for {\em some} $\eps_0$ & Single & Triangle Detection hypothesis\\
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$\Omega\inparen{\inparen{\qruntime{Q, \dbbase}}^{1+\eps_0}}$ for {\em some} $\eps_0>0$ & Single & Triangle Detection hypothesis\\
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\hline
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$\omega\inparen{\inparen{\qruntime{Q, \dbbase}}^{C_0}}$ for {\em all} $C_0>0$ & Multiple &$\sharpwzero\ne\sharpwone$\\
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\hline
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$\Omega\inparen{\inparen{\qruntime{Q, \dbbase}}^{c_0\cdot k}}$ for {\em some} $c_0>0$ & Multiple & Current algorithms for counting $k$-matchings\\
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\hline
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\hline
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$(\qruntime{Q, \dbbase})^{c_0}$ for {\em all} $\eps_0$ & Multiple & Triangle Detection hypothesis\\
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\end{tabular}
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\end{tabular}
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\caption{Our lower bounds for a specific hard query $Q$ parameterized by $k$. The $\pdb$ is over the same $\dbbase$ and those with `Multiple' in the second column need the algorithm to be able to handle multiple $\pd$. The last column states the hardness assumptions that imply the lower bounds in the first column (all of $\eps_o,C_0,c_0$ are all constants independent of $k$).}
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\label{tab:lbs}
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\end{table}
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\end{table}
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Note that the lower bound in the first row by itself is enough to refute \Cref{prob:informal}.
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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 tuple $\tup$ (which is the parameter that defines our family of hard queries). What our lower bound in the third rows says that one cannot get more than a polynomial improvement over essentially the trivial algorithm for \Cref{prob:informal}. 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). To put these hardness results in context, we will next take a short detour to review the existing hardness results for \abbrPDB\xplural under set semantics.
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% Atri: Converting sub-section to para since it saves space
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% Atri: Converting sub-section to para since it saves space
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@ -293,6 +293,7 @@
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\newcommand{\sharpp}{\#{\sf P}\xspace}
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\newcommand{\sharpp}{\#{\sf P}\xspace}
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\newcommand{\sharpphard}{\#{\sf P}-hard\xspace}
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\newcommand{\sharpphard}{\#{\sf P}-hard\xspace}
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\newcommand{\sharpwone}{\#{\sf W}[1]\xspace}
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\newcommand{\sharpwone}{\#{\sf W}[1]\xspace}
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\newcommand{\sharpwzero}{\#{\sf W}[0]\xspace}
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\newcommand{\sharpwonehard}{\#{\sf W}[1]-hard\xspace}
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\newcommand{\sharpwonehard}{\#{\sf W}[1]-hard\xspace}
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\newcommand{\ptime}{{\sf PTIME}\xspace}
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\newcommand{\ptime}{{\sf PTIME}\xspace}
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\newcommand{\timeOf}[1]{T_{#1}}
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\newcommand{\timeOf}[1]{T_{#1}}
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