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\label{sec:hard}
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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.
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Note that this implies hardness for \bis and general \abbrBPDB, answering \Cref{prob:bag-pdb-poly-expected}
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Note that this implies hardness for \bis and general \abbrBPDB, showing \Cref{prob:bag-pdb-poly-expected} cannot be done in $O\inparen{\qruntime{\query,\dbbase}}$ runtime.
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%(and hence the equivalent \Cref{prob:bag-pdb-query-eval})
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in the negative.
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%in the negative.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Preliminaries}\label{sec:hard:sub:pre}
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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.
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