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#CFG and #DNNF admit FPRAS

#CFG and #DNNF admit FPRAS.
Kuldeep S. Meel, and Alexis de Colnet.
In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pp. 5978–6010, 2026.

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Abstract

We provide the first fully polynomial-time randomized approximation scheme for the following two counting problems: 1. Given a Context-Free Grammar G over alphabet Sigma , count the number of words of length exactly n generated by G 2. Given a circuit phi in Decomposable Negation Normal Form (DNNF) over the set of Boolean variables X , compute the number of assignments to X such that phi evaluates to 1. Finding polynomial time algorithms for the aforementioned problems has been a longstanding open problem. Prior work could either only obtain a quasi-polynomial runtime or a polynomial-time randomized approximation scheme for non-deterministic finite automata and non-deterministic tree automata.

BibTeX

@inproceedings{MdC26,
  title={\#CFG and \#DNNF admit FPRAS},
  author={Meel, Kuldeep S. and de Colnet, Alexis},
  booktitle={
    Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)
  },
  pages={5978--6010},
  year={2026},
  bib2html_rescat={Counting},
  bib2html_pubtype={Refereed Conference},
  bib2html_dl_pdf={https://doi.org/10.1137/1.9781611978971.213},
  abstract={
    We provide the first fully polynomial-time randomized approximation scheme
    for the following two counting problems:
    1.
    Given a Context-Free Grammar G
    over alphabet Sigma
    , count the number of words of length exactly n
    generated by G
    2.
    Given a circuit phi
    in Decomposable Negation Normal Form (DNNF) over the set of Boolean
    variables X
    , compute the number of assignments to X
    such that phi
    evaluates to 1.
    Finding polynomial time algorithms for the aforementioned problems has been
    a longstanding open
    problem. Prior work could either only obtain a quasi-polynomial runtime or a
    polynomial-time
    randomized approximation scheme for non-deterministic finite automata and
    non-deterministic
    tree automata.
  },
}

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