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@inproceedings{DM19,
  title={Dual Hashing-based Algorithms for Discrete Integration},
  author={de Colnet,Alexis and Meel, Kuldeep S.},
  year={2019},
  month=oct,
  booktitle=CP,
  bib2html_rescat={Counting},
  bib2html_dl_pdf={../Papers/cp19-dm.pdf},
  bib2html_pubtype={Refereed Conference},
  abstract={
    Given a boolean formula F and a weight function $\rho$, the problem of
    discrete integration seeks to compute the weight of $F$, defined as the sum
    of the weights of satisfying assignments. Discrete integration, also known
    as weighted model counting, is a fundamental problem in computer science
    with wide variety of applications ranging from machine learning and
    statistics to physics and infrastructure reliability. Given the
    intractability of the exact variant, the problem of approximate weighted
    model counting has been subject to intense theoretical and practical
    investigations over the years.
    The primary contribution of this paper is to investigate development of
    algorithmic approaches for discrete integration. Our framework allows us to
    derive two different algorithms, which can be seen as dual to each other:
    \textsf{WISH}, which was already discovered by Ermon et al~\cite{EGSS13c},
    and a new algorithm: \textsf{SWITCH}.We argue that these algorithms can be
    seen as dual to each other, in the sense that their complexities differ only
    by a permutation of certain parameters. Indeed we show that, for $F$ defined
    over $n$ variables, a weight function $\rho$ that can be represented using
    $p$ bits, and a confidence parameter $\delta$, there is a function $f$ and
    an NP oracle such that \textsf{WISH} makes $\mathcal{O}
    \left(f(n,p,\delta)\right)$ calls to NP oracle while \textsf{SWITCH} makes
    $\mathcal{O}\left(f(p,n,\delta)\right)$ queries. We find $f(x,y,\delta)$
    polynomial in $x$, $y$ and $1/\delta$, more specifically $f(x,y,\delta) =
    x\log(y)\log(x/\delta)$.
    We first focus on striking similarities of both the design process and
    structure of the two algorithms but then show that despite this
    quasi-symmetry, the analysis yields time complexities dual to each other.
    Another contribution of this paper is the use of 3-wise property
    independence of XOR based hash functions in the analysis of WISH and SWITCH.
    To the best of our knowledge, this is the first usage of 3-wise independence
    in deriving stronger concentration bounds.
  },
}