Update of References in main.bib
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@ -29,7 +29,7 @@ We call a polynomial $\poly\inparen{\vct{X}}$ a \emph{\abbrCTIDB-lineage polynom
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\noindent A block independent database \abbrBIDB $\pdb'$ models a set of worlds each of which consists of a subset of the possible tuples $\tupset'$, where $\tupset'$ is partitioned into $\numblock$ blocks $\block_i$ and all $\block_i$ are independent random events. $\pdb'$ further constrains that all $\tup\in\block_i$ for all $i\in\pbox{\numblock}$ of $\tupset'$ be disjoint events. We refer to any monomial that includes $X_\tup X_{\tup'}$ for $\tup\neq\tup'\in\block_i$ as a \emph{cancellation}. We define next a specific construction of \abbrBIDB that is useful for our work.
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\begin{Definition}[\abbrOneBIDB]\label{def:one-bidb}
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Define a \emph{\abbrOneBIDB} to be the pair $\pdb' = \inparen{\bigtimes_{\tup\in\tupset'}\inset{0, \bound_\tup}, \bpd'},$ where $\tupset'$ is the set of possible tuples such that each $\tup \in \tupset'$ has a multiplicity domain of $\inset{0, \bound_\tup}$, with $\bound_\tup\in\mathbb{N}$. $\tupset'$ is partitioned into $\numblock$ independent blocks $\block_i,$ for $i\in\pbox{\numblock}$, of disjoint tuples. $\bpd'$ is characterized by the vector $\inparen{\prob_\tup}_{\tup\in\tupset'}$ where for every block $\block_i$, $\sum_{\tup \in \block_i}\prob_\tup \leq 1$. Given $W\in\onebidbworlds{\tupset'}$ and for $i\in\pbox{\numblock}$, let $\prob_i(W) = \begin{cases}
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Define a \emph{\abbrOneBIDB} to be the pair $\pdb' = \inparen{\bigtimes_{\tup\in\tupset'}\inset{0, \bound_\tup}, \bpd'},$ where $\tupset'$ is the set of possible tuples such that each $\tup \in \tupset'$ has a multiplicity domain of $\inset{0, \bound_\tup}$, with $\bound_\tup\in\mathbb{N}$. $\tupset'$ is partitioned into $\numblock$ independent blocks $\block_i,$ for $i\in\pbox{\numblock}$, of disjoint tuples. $\bpd'$ is characterized by the vector $\inparen{\prob_\tup}_{\tup\in\tupset'}$ where for every block $\block_i$, $\sum_{\tup \in \block_i}\prob_\tup \leq 1$. Given $W\in\onebidbworlds{\tupset'}$ and for $i\in\pbox{\numblock}$, let $\prob_\tup(W) = \begin{cases}
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1 - \sum_{\tup\in\block_i}\prob_\tup & \text{if }W_\tup = 0\text{ for all }\tup\in\block_i\\
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0 & \text{if there exists } \tup \neq \tup'\in\block_i; W_\tup, W_{\tup'}\neq 0\\
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\prob_\tup & W_\tup \ne 0 \text{ for the unique } t\in B_i.\\
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@ -122,7 +122,7 @@ Further, our approximation algorithm works for a more general notion of bag \abb
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(see \Cref{subsec:tidbs-and-bidbs}).
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\subsection{Polynomial Equivalence}\label{sec:intro-poly-equiv}
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A common encoding of probabilistic databases (e.g., in \cite{IL84a,Imielinski1989IncompleteII,4497507,DBLP:conf/vldb/AgrawalBSHNSW06} and many others) annotates tuples with lineages, propositional formulas that describe the set of possible worlds that the tuple appears in. The bag semantics analog is a provenance/lineage polynomial (see~\Cref{fig:nxDBSemantics}) $\apolyqdt$~\cite{DBLP:conf/pods/GreenKT07}, a polynomial with non-zero integer coefficients and exponents, over variables $\vct{X}$ encoding input tuple multiplicities. The lineage polynomial for result tuple $t_{out}$ evaluates to $t_{out}$'s multiplicity in a given possible world when each $X_{t_{in}}$ is replaced by the multiplicity of $t_{in}$ in the possible world.
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A common encoding of probabilistic databases (e.g., in \cite{IL84a,4497507,DBLP:conf/vldb/AgrawalBSHNSW06} and many others) annotates tuples with lineages, propositional formulas that describe the set of possible worlds that the tuple appears in. The bag semantics analog is a provenance/lineage polynomial (see~\Cref{fig:nxDBSemantics}) $\apolyqdt$~\cite{DBLP:conf/pods/GreenKT07}, a polynomial with non-zero integer coefficients and exponents, over variables $\vct{X}$ encoding input tuple multiplicities. The lineage polynomial for result tuple $t_{out}$ evaluates to $t_{out}$'s multiplicity in a given possible world when each $X_{t_{in}}$ is replaced by the multiplicity of $t_{in}$ in the possible world.
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We drop $\query$, $\tupset$, and $\tup$ from $\apolyqdt$ when they are clear from the context or irrelevant to the discussion. We now specify the problem of computing the expectation of tuple multiplicity in the language of lineage polynomials:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -187,7 +187,7 @@ We have argued that for our specific example the expectation that we want is $\r
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For any \abbrCTIDB $\pdb$, $\raPlus$ query $\query$, and lineage polynomial
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$\poly\inparen{\vct{X}}=\poly\pbox{\query,\tupset,\tup}\inparen{\vct{X}}$, it holds that $
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\expct_{\vct{W} \sim \pdassign}\pbox{\poly\inparen{\vct{W}}} = \rpoly\inparen{\probAllTup}
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$, where $\probAllTup = \inparen{\prob_{\tup}}_{\tup\in\tupset}.$
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$, where $\probAllTup = \inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}.$
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\end{Lemma}
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\noindent
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50
main.bbl
50
main.bbl
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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}{51}
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\begin{thebibliography}{50}
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%%% ====================================================================
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%%% NOTE TO THE USER: you can override these defaults by providing
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@ -104,8 +104,8 @@
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Frohm}, \bibinfo{person}{Charles~M. Gaona}, \bibinfo{person}{Gary~D.
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Hachtel}, \bibinfo{person}{Enrico Macii}, \bibinfo{person}{Abelardo Pardo},
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{and} \bibinfo{person}{Fabio Somenzi}.} \bibinfo{year}{1993}\natexlab{}.
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\newblock \showarticletitle{Algebraic decision diagrams and their
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applications}. In \bibinfo{booktitle}{\emph{IEEE CAD}}.
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\newblock \showarticletitle{Algebraic Decision Diagrams and Their
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Applications}. In \bibinfo{booktitle}{\emph{IEEE CAD}}.
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\newblock
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@ -171,19 +171,20 @@
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\bibitem[Curticapean and Marx(2014)]%
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{DBLP:journals/corr/CurticapeanM14}
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{10.1109/FOCS.2014.22}
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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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\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}. In
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\bibinfo{booktitle}{\emph{Proceedings of the 2014 IEEE 55th Annual Symposium
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on Foundations of Computer Science}} \emph{(\bibinfo{series}{FOCS '14})}.
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\bibinfo{publisher}{IEEE Computer Society}, \bibinfo{address}{USA},
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\bibinfo{pages}{130–139}.
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\newblock
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\showeprint[arXiv]{1407.2929}
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\showISBNx{9781479965175}
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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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\url{https://doi.org/10.1109/FOCS.2014.22}
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\showDOI{\tempurl}
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\bibitem[Dalvi and Suciu(2007a)]%
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@ -309,15 +310,12 @@
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\bibitem[Flum and Grohe(2006)]%
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{param-comp}
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\bibfield{author}{\bibinfo{person}{J{\"{o}}rg Flum} {and}
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\bibinfo{person}{Martin Grohe}.} \bibinfo{year}{2006}\natexlab{}.
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\newblock \bibinfo{booktitle}{\emph{Parameterized Complexity Theory}}.
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\newblock \bibinfo{publisher}{Springer}.
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\bibfield{author}{\bibinfo{person}{J{\"o}rg Flum} {and} \bibinfo{person}{Martin
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Grohe}.} \bibinfo{year}{2006}\natexlab{}.
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\newblock \showarticletitle{Parameterized Complexity Theory}. In
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\bibinfo{booktitle}{\emph{Texts in Theoretical Computer Science. An EATCS
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Series}}.
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\newblock
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\showISBNx{978-3-540-29952-3}
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\urldef\tempurl%
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\url{https://doi.org/10.1007/3-540-29953-X}
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\showDOI{\tempurl}
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\bibitem[Garcia{-}Molina et~al\mbox{.}(2009)]%
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@ -325,7 +323,7 @@
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\bibfield{author}{\bibinfo{person}{Hector Garcia{-}Molina},
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\bibinfo{person}{Jeffrey~D. Ullman}, {and} \bibinfo{person}{Jennifer Widom}.}
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\bibinfo{year}{2009}\natexlab{}.
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\newblock \bibinfo{booktitle}{\emph{Database systems - the complete book {(2.}
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\newblock \bibinfo{booktitle}{\emph{Database Systems - The Complete Book {(2.}
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ed.)}}.
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\newblock \bibinfo{publisher}{Pearson Education}.
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\newblock
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@ -364,14 +362,6 @@
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\showDOI{\tempurl}
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\bibitem[Imielinski and Lipski(1989)]%
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{Imielinski1989IncompleteII}
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\bibfield{author}{\bibinfo{person}{T. Imielinski} {and} \bibinfo{person}{W.
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Lipski}.} \bibinfo{year}{1989}\natexlab{}.
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\newblock \showarticletitle{Incomplete Information in Relational Databases}.
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\newblock
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\bibitem[Imieli\'nski and Lipski~Jr(1984)]%
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{IL84a}
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\bibfield{author}{\bibinfo{person}{Tomasz Imieli\'nski} {and}
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58
main.bib
58
main.bib
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@ -1,17 +1,17 @@
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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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@inproceedings{10.1109/FOCS.2014.22,
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author = {Curticapean, Radu and Marx, D\'{a}niel},
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title = {Complexity of Counting Subgraphs: Only the Boundedness of the Vertex-Cover Number Counts},
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year = {2014},
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isbn = {9781479965175},
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publisher = {IEEE Computer Society},
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address = {USA},
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url = {https://doi.org/10.1109/FOCS.2014.22},
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doi = {10.1109/FOCS.2014.22},
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abstract = {For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertex-cover number (equivalently, the size of the maximum matching in C is bounded), then #Sub(C) is polynomial-time solvable. We complement this result with a corresponding lower bound: if C is any recursively enumerable class of graphs with unbounded vertex-cover number, then #Sub(C) is #W[1]-hard parameterized by the size of H and hence not polynomial-time solvable and not even fixed-parameter tractable, unless FPT is equal to #W[1]. As a first step of the proof, we show that counting k-matchings in bipartite graphs is #W[1]-hard. Recently, Curticapean [ICALP 2013] proved the #W[1]-hardness of counting k-matchings in general graphs, our result strengthens this statement to bipartite graphs with a considerably simpler proof and even shows that, assuming the Exponential Time Hypothesis (ETH), there is no f(k)*no(k/log(k)) time algorithm for counting k-matchings in bipartite graphs for any computable function f. As a consequence, we obtain an independent and somewhat simpler proof of the classical result of Flum and Grohe [SICOMP 2004] stating that counting paths of length k is #W[1]-hard, as well as a similar almost-tight ETH-based lower bound on the exponent.},
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booktitle = {Proceedings of the 2014 IEEE 55th Annual Symposium on Foundations of Computer Science},
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pages = {130–139},
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numpages = {10},
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series = {FOCS '14}
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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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@ -79,12 +79,6 @@ series = {FOCS '02}
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year = {2018}
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}
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@inproceedings{Imielinski1989IncompleteII,
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title={Incomplete Information in Relational Databases},
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author={T. Imielinski and W. Lipski},
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year={1989}
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}
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@inproceedings{10.1145/1265530.1265571,
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author = {Dalvi, Nilesh and Suciu, Dan},
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booktitle = {PODS},
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@ -561,7 +555,7 @@ Virginia Vassilevska Williams},
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D. Hachtel and Enrico Macii and Abelardo Pardo and Fabio
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Somenzi},
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booktitle = {IEEE CAD},
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title = {Algebraic decision diagrams and their applications},
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title = {Algebraic Decision Diagrams and Their Applications},
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year = {1993}
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}
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@ -635,25 +629,19 @@ Maximilian Schleich},
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bibsource = {dblp computer science bibliography, https://dblp.org}
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}
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@book{param-comp,
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author = {J{\"{o}}rg Flum and
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Martin Grohe},
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title = {Parameterized Complexity Theory},
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series = {Texts in Theoretical Computer Science. An {EATCS} Series},
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publisher = {Springer},
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year = {2006},
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url = {https://doi.org/10.1007/3-540-29953-X},
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doi = {10.1007/3-540-29953-X},
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isbn = {978-3-540-29952-3},
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timestamp = {Tue, 16 May 2017 14:24:38 +0200},
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biburl = {https://dblp.org/rec/series/txtcs/FlumG06.bib},
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bibsource = {dblp computer science bibliography, https://dblp.org}
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@inproceedings{param-comp,
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title={Parameterized Complexity Theory},
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author={J{\"o}rg Flum and Martin Grohe},
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booktitle={Texts in Theoretical Computer Science. An EATCS Series},
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year={2006}
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}
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@book{DBLP:books/daglib/0020812,
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author = {Hector Garcia{-}Molina and
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Jeffrey D. Ullman and
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Jennifer Widom},
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title = {Database systems - the complete book {(2.} ed.)},
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title = {Database Systems - The Complete Book {(2.} ed.)},
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publisher = {Pearson Education},
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year = {2009}
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}
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@ -19,7 +19,7 @@ Given positive integer $k$ and undirected graph $G=(\vset,\edgeSet)$ with no sel
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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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\begin{hypo}[~\cite{10.1109/FOCS.2014.22}]\label{conj:known-algo-kmatch}
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For every $G=\inparen{\vset, \edgeSet}$, $\kmatchtime\ge n^{\Omega\inparen{k/\log{k}}}$.
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\end{hypo}
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