Changes to Conjecture 3.2.
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@ -82,6 +82,14 @@ Denote the vector $\vct{p}$ to be a vector whose elements are the individual pro
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Recall that tuple blocks in a TIDB always have size 1, so the outer summation of \cref{eq:tidb-expectation} is over the full set of vectors.
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\subsection{Proof of~\Cref{prop:ctidb-reduct}}
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\begin{proof}[Proof of~\Cref{prop:ctidb-reduct}]
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We first need to prove that any \abbrCTIDB $\pdb$ can be reduced to the \abbrOneBIDB created by~\Cref{prop:ctidb-reduct}. By definition, any $\tup\in\tupset$ can be present $c'\in\pbox{\bound}$ times in the possible worlds it appears in, with a disjoint probability distribution across the multiplicities $\pbox{\bound}$. Then the construction of~\Cref{prop:ctidb-reduct} of the block of tuples $\inset{\intuple{\tup, j}_{j\in\pbox{\bound}}}$ for each $\tup\in\tupset$ indeed encodes the disjoint behavior across multiplicities. $\pdb$ further requires that all $\tup\in\tupset$ are independent, a property which is enforced by independence constraint across all blocks of tuples in~\Cref{def:one-bidb}. Then the construction of $\pdb'$ in~\Cref{prop:ctidb-reduct} is an equivalent representation of $\pdb$.
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Next we need to show that the distributions over $\pdb$ and $\pdb'$ are equivalent. The distribution $\bpd$ is a distribution disjoint across the set of multiplicities $\pbox{\bound}$ and independent across all $\tup\in\tupset$. By definition,~\Cref{prop:ctidb-reduct} creates $\pdb'$ by producing a block of disjoint tuples $\tup_j = \intuple{\tup, j}$ for each $\tup\in\tupset$ and $j\in\pbox{\bound}$, where $\vct{\prob} = \inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}$. Since the probability vector $\vct{\prob_{\pdb}} = \inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}$ \emph{and} each $\prob_{\tup, j}, \prob_{\tup, j'}$ for $j\neq j'\in\pbox{\bound}$ are disjoint, the distributions are hence the same.
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\end{proof}
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\subsection{Proof of~\Cref{prop:expection-of-polynom}}
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\label{subsec:expectation-of-polynom-proof}
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\begin{proof}
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@ -127,18 +135,17 @@ Note that any $\poly$ in factorized form is equivalent to its \abbrSMB expansion
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\subsection{Proof for Lemma~\ref{lem:tidb-reduce-poly} and~\Cref{lem:bin-bidb-phi-eq-redphi}}\label{subsec:proof-exp-poly-rpoly}
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\begin{proof}
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Let $\poly$ be a polynomial of $\numvar$ variables with highest degree $= \hideg$, defined as follows:
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\[\poly(X_1,\ldots, X_\numvar) = \sum_{\vct{d} \in \{0,\ldots, \hideg\}^\numvar}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar X_i^{d_i}.\]
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\[\poly(X_1,\ldots, X_\numvar) = \sum_{\vct{d} \in \{0,\ldots, \hideg\}^\numvar}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar X_i^{d_i},\]
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where $\tupset'$ has $\numvar$ tuples, we can equivalently write $\prod\limits_{\substack{\tup\in\tupset'\\s.t.~d_\tup\geq 1}}$ for the product term.
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%If we can prove that $\poly\inparen{\vct{W}} = \rpoly\inparen{\vct{W}}$ for any $\poly\inparen{\vct{X}}$ and any $\vct{W}$, then the proof holds for any $\refpoly{}\inparen{\vct{W}}$ since $\refpoly{}\inparen{\vct{W}}$ is \emph{itself} a polynomial as defined above.\footnote{This can be seen in converting $\refpoly{}\inparen{\vct{X}}$ into \abbrSMB.}
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Let the boolean function $\isInd{\cdot}$ take $\vct{d}$ as input and return true if there does not exist any dependent variables in $\vct{d}$, i.e., for block $\block$ and $i, j\in\pbox{\numvar}$, $\not\exists ~\block, i\neq j\suchthat \block\supseteq\inset{\tup_i, \tup_j} \wedge d_{i}, d_{j} \geq 1$.
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Let the boolean function $\isInd{\cdot}$ take $\vct{d}$ as input and return true if there does not exist any dependent variables in the monomial encoded by $\vct{d}$, i.e., for any block $\block\in\tupset'$, $\not\exists \tup, \tup' \in\block~|~\vct{d}_\tup, \vct{d}_{\tup'}\geq 1$.% \and $\tup, \tup'\in\tupset\implies \inparen{\vct{d}_t=0\vee\vct{d}_{\tup'} = 0}\$, $\not\exists ~\block, i\neq j\suchthat \block\supseteq\inset{\tup_i, \tup_j} \wedge d_{i}, d_{j} \geq 1$.
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%\footnote{For generality of the proof, we are using a slightly different notation than the main paper, which treats a specific form of \abbrBIDB} For clarity, a \abbrOneBIDB polynomial $\poly\inparen{\vct{X}}$ with any variable $X_i$ such that $X_i\in\inset{0, \bound_\tup}, \bound_\tup\neq 1$ can equivalently replace $X_i$ with $\bound_\tup X_i$ while coercing the domain of $X_i$ to be $\inset{0, 1}$. Note that this setup addresses the general \abbrBIDB. In what follows, we assume that $\vct{X}$ (and hence $\vct{W}$) has a domain of $\inset{0, 1}$.
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Note that when computing the expectation of a \abbrOneBIDB polynomial over the set of worlds, any world $\vct{W}\in \bigtimes_{\tup \in \tupset}\inset{0, \bound_\tup}$ with $W_\tup \geq 2$ for $W_\tup\in\vct{W}$ can equivalently be replaced with $\bound_\tup^{d_t}\cdot W_\tup$ with domain of $W_t$ $\domain\inparen{W_\tup} = \inset{0, 1}$. In what follows, we replace all such $W_\tup$ with their equivalents and without loss of generality assume that $c_{\vct{d}}$ has absorbed all such $\bound_\tup^{d_\tup}$ factors.
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Then, given \abbrOneBIDB $\pdb$, query $\query$, and polynomial $\poly\inparen{\vct{W}} = \poly\pbox{\query, \tupset, \tup}$, in expectation we have
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\AH{Two considerations:\begin{itemize}\item I suggest that we change the domain of $\vct{d}$ to $\vct{d}\in\inset{1,\cdots\hideg}$.\item It seems `bulky' to try and use the paper's conventional $\tup_{i, j}$ notation in this proof, since the proof is for general \abbrOneBIDB; therefore the notation is cleaner to iterate through $\pbox{\numvar}$ imo.\end{itemize}
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}
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\begin{align}
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\expct_{\vct{\randWorld}}\pbox{\poly(\vct{\randWorld})} &= \expct_{\vct{\randWorld}}\pbox{\sum_{\substack{\vct{d} \in \{0,\ldots,\hideg\}^\numvar\\\wedge~\isInd{\vct{d}}}}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \randWorld_i^{d_i} + \sum_{\substack{\vct{d} \in \{0,\ldots, \hideg\}^\numvar\\\wedge ~\neg\isInd{\vct{d}}}} c_{\vct{d}}\cdot\prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar\randWorld_i^{d_i}}\label{p1-s1a}\\
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&= \sum_{\substack{\vct{d} \in \{0,\ldots,\hideg\}^\numvar\\\wedge~\isInd{\vct{d}}}}c_{\vct{d}}\cdot \expct_{\vct{\randWorld}}\pbox{\prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \randWorld_i^{d_i}} + \sum_{\substack{\vct{d} \in \{0,\ldots, \hideg\}^\numvar\\\wedge ~\neg\isInd{\vct{d}}}} c_{\vct{d}}\cdot\expct_{\vct{\randWorld}}\pbox{\prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar\randWorld_i^{d_i}}\label{p1-s1b}\\
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@ -148,7 +155,7 @@ Then, given \abbrOneBIDB $\pdb$, query $\query$, and polynomial $\poly\inparen{\
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&= \sum_{\substack{\vct{d} \in \{0,\ldots,\hideg\}^\numvar\\\wedge~\isInd{\vct{d}}}}c_{\vct{d}}\cdot \prod_{\substack{i = 1\\s.t. d_i \geq 1}}^\numvar \prob_i\label{p1-s4}\\
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&= \rpoly(\prob_1,\ldots, \prob_\numvar).\label{p1-s5}
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\end{align}
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\Cref{p1-s1a} is the result of substituting in the definition of $\poly$ given above. Then we arrive at \cref{p1-s1b} by linearity of expectation. Next, \cref{p1-s1c} is the result of the independence constraint of \abbrBIDB\xplural, specifically that any monomial composed of dependent variables, i.e., variables from the same block $\block$, has a probability of $0$. \Cref{p1-s2} is obtained by the fact that all variables in each monomial are independent, which allows for the expectation to be pushed through the product. In \cref{p1-s3}, since $\randWorld_i \in \{0, 1\}$ it is the case that for any exponent $e \geq 1$, $\randWorld_i^e = \randWorld_i$. Next, in \cref{p1-s4} the expectation of a tuple is indeed its probability.
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\Cref{p1-s1a} is the result of substituting in the definition of $\poly$ given above. Then we arrive at \cref{p1-s1b} by linearity of expectation. Next, \cref{p1-s1c} is the result of the independence constraint of \abbrBIDB\xplural, specifically that any monomial composed of dependent variables, i.e., variables from the same block $\block$, has a probability of $0$. \Cref{p1-s2} is obtained by the fact that all variables in each monomial are independent, which allows for the expectation to be pushed through the product. In \cref{p1-s3}, note that when computing the expectation of a \abbrOneBIDB polynomial over the set of worlds, any world $\vct{W}\in \bigtimes_{\tup \in \tupset}\inset{0, \bound_\tup}$ with $W_\tup \geq 2$ for $W_\tup\in\vct{W}$ can equivalently be replaced with $\bound_\tup^{d_t}\cdot W_\tup$ with domain of $W_t$ $\domain\inparen{W_\tup} = \inset{0, 1}$. Then $c_{\vct{d}}$ absorbs all such $\bound_\tup^{d_\tup}$ factors. Since $\randWorld_i \in \{0, 1\}$ it is the case that for any exponent $e \geq 1$, $\randWorld_i^e = \randWorld_i$. Next, in \cref{p1-s4} the expectation of a tuple is indeed its probability.
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Finally, it can be verified that \Cref{p1-s5} follows since \cref{p1-s4} satisfies the construction of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ in \Cref{def:reduced-poly}.
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\qed
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@ -59,14 +59,6 @@ Given \abbrCTIDB $\pdb =$\newline $\inparen{\worlds, \bpd}$, let $\pdb' = \inpar
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\end{Proposition}
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\AH{Not entirely sure if I the notation above $\bound_{\tup_j} = j$ for each $\tup_j$ in $\tupset'$.}
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{\color{red}
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\begin{proof}[Proof of~\Cref{prop:ctidb-reduct}]
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We first need to prove that any \abbrCTIDB $\pdb$ can be reduced to the \abbrOneBIDB created by~\Cref{prop:ctidb-reduct}. By definition, any $\tup\in\tupset$ can be present $c'\in\pbox{\bound}$ times in the possible worlds it appears in, with a disjoint probability distribution across the multiplicities $\pbox{\bound}$. Then the construction of~\Cref{prop:ctidb-reduct} of the block of tuples $\inset{\intuple{\tup, j}_{j\in\pbox{\bound}}}$ for each $\tup\in\tupset$ indeed encodes the disjoint behavior across multiplicities. $\pdb$ further requires that all $\tup\in\tupset$ are independent, a property which is enforced by independence constraint across all blocks of tuples in~\Cref{def:one-bidb}. Then the construction of $\pdb'$ in~\Cref{prop:ctidb-reduct} is an equivalent representation of $\pdb$.
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Next we need to show that the distributions over $\pdb$ and $\pdb'$ are equivalent. The distribution $\bpd$ is a distribution disjoint across the set of multiplicities $\pbox{\bound}$ and independent across all $\tup\in\tupset$. By definition,~\Cref{prop:ctidb-reduct} creates $\pdb'$ by producing a block of disjoint tuples $\tup_j = \intuple{\tup, j}$ for each $\tup\in\tupset$ and $j\in\pbox{\bound}$, where $\vct{\prob} = \inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}$. Since the probability vector $\vct{\prob_{\pdb}} = \inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}$ \emph{and} each $\prob_{\tup, j}, \prob_{\tup, j'}$ for $j\neq j'\in\pbox{\bound}$ are disjoint, the distributions are hence the same.
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\end{proof}
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}
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We now define the reduced polynomial $\rpoly'$ of a \abbrOneBIDB.
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\begin{figure}[t!]
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%\centering
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@ -261,7 +261,7 @@ Bags, as we consider, are sufficient for production use, where bag-relational al
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Our results show that bag-\abbrPDB\xplural can be competitive, laying the groundwork for probabilistic functionality in production database engines.
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\mypar{Concurrent Work}
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In work unpublished and independent of ours, Grohe, et. al.~\cite{https://doi.org/10.48550/arxiv.2201.11524} investigate \abbrBPDB\xplural showing a dichotomy for the query evaluation problem over \abbrBPDB\xplural when computing the probability of an output tuple having at most a multiplicity of $k$. This work allows for unbounded multiplicities and therefore addresses the concern of a succinct representation of the distribution over infinitely many multiplicities. Our work in contrast assumes a finite bound on the multiplicities where we simply list the finitely many probability values. The work also observes that computing the expected value of an output tuple multiplicity is in polynomial time, but does not, as this work does, further look into the fine-grained analysis of computing this value.
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In work independent of ours, Grohe, et. al.~\cite{https://doi.org/10.48550/arxiv.2201.11524} investigate bag-\abbrTIDB\xplural allowing for unbounded multiplicities and therefore addressing the concern of a succinct representation of the distribution over infinitely many multiplicities. While the authors observe that computing the expected value of an output tuple multiplicity is in polynomial time, no further analysis of the expected value is considered. The work investigates the query evaluation problem over bag-\abbrTIDB\xplural when computing the probability of an output tuple having at most a multiplicity of $k$, showing that a dichotomy exists for this problem. Our work in contrast assumes a finite bound on the multiplicities where we simply list the finitely many probability values and further looks into the fine-grained analysis of computing the expected multiplicity of an output tuple..
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18
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18
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@ -2,7 +2,7 @@
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%%% Do NOT edit. File created by BibTeX with style
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%%% ACM-Reference-Format-Journals [18-Jan-2012].
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\begin{thebibliography}{50}
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\begin{thebibliography}{51}
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%%% ====================================================================
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%%% NOTE TO THE USER: you can override these defaults by providing
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@ -170,6 +170,22 @@
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\newblock
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\bibitem[Curticapean and Marx(2014)]%
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{DBLP:journals/corr/CurticapeanM14}
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\bibfield{author}{\bibinfo{person}{Radu Curticapean} {and}
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\bibinfo{person}{D{\'{a}}niel Marx}.} \bibinfo{year}{2014}\natexlab{}.
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\newblock \showarticletitle{Complexity of counting subgraphs: only the
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boundedness of the vertex-cover number counts}.
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\newblock \bibinfo{journal}{\emph{CoRR}} \bibinfo{volume}{abs/1407.2929}
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(\bibinfo{year}{2014}).
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\newblock
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\showeprint[arXiv]{1407.2929}
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\urldef\tempurl%
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\url{http://arxiv.org/abs/1407.2929}
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\showURL{%
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\tempurl}
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\bibitem[Dalvi and Suciu(2007a)]%
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{10.1145/1265530.1265571}
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\bibfield{author}{\bibinfo{person}{Nilesh Dalvi} {and} \bibinfo{person}{Dan
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15
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15
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@ -1,3 +1,18 @@
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@article{DBLP:journals/corr/CurticapeanM14,
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author = {Radu Curticapean and
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D{\'{a}}niel Marx},
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title = {Complexity of counting subgraphs: only the boundedness of the vertex-cover
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number counts},
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journal = {CoRR},
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volume = {abs/1407.2929},
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year = {2014},
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url = {http://arxiv.org/abs/1407.2929},
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eprinttype = {arXiv},
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eprint = {1407.2929},
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timestamp = {Mon, 13 Aug 2018 16:48:39 +0200},
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biburl = {https://dblp.org/rec/journals/corr/CurticapeanM14.bib},
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bibsource = {dblp computer science bibliography, https://dblp.org}
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}
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@misc{https://doi.org/10.48550/arxiv.2201.11524,
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doi = {10.48550/ARXIV.2201.11524},
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url = {https://arxiv.org/abs/2201.11524},
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@ -15,10 +15,17 @@ In particular, we will consider the problems of computing the following counts (
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Given positive integer $k$ and undirected graph $G=(\vset,\edgeSet)$ with no self-loops or parallel edges, $\kmatchtime\ge \littleomega{f(k)\cdot |\edgeSet|^c}$ for any function $f$ and any constant $c$ independent of $\abs{E}$ and $k$ (assuming $\sharpwzero\ne\sharpwone$).
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\end{Theorem}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{hypo}\label{conj:known-algo-kmatch}
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There exists an absolute constant $c_0>0$ such that for every $G=(\vset,\edgeSet)$, we have $\kmatchtime \ge \Omega\inparen{|E|^{c_0\cdot k}}$ for large enough $k$.
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%\begin{hypo}\label{conj:known-algo-kmatch}
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%There exists an absolute constant $c_0>0$ such that for every $G=(\vset,\edgeSet)$, we have $\kmatchtime \ge \Omega\inparen{|E|^{c_0\cdot k}}$ for large enough $k$.
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%\end{hypo}
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\begin{hypo}[~\cite{DBLP:journals/corr/CurticapeanM14}]\label{conj:known-algo-kmatch}
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For every $G=\inparen{\vset, \edgeSet}$, we have $\kmatchtime\ge n^{\Omega\inparen{k/\log{k}}}$.
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\end{hypo}
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We note that the above conjecture is somewhat non-standard. In particular, the best known algorithm to compute $\numocc{G}{\kmatch}$ takes time $\Omega\inparen{|V|^{k/2}}$ (i.e. if this is the best algorithm then $c_0=\frac 14$)~\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 $\numocc{G}{\kmatch}$.
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We note that the above conjecture is somewhat non-standard. In particular, the best known algorithm to compute $\numocc{G}{\kmatch}$ takes time $\Omega\inparen{|V|^{k/2}}$
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%(i.e. if this is the best algorithm then $c_0=\frac 14$)
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~\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 $\numocc{G}{\kmatch}$.
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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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Loading…
Reference in New Issue