Efficient computation of representative weight functions with applications to parameterized counting (extended version)

Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

In this paper we prove an analogue of the classic Bollobás lemma for approximate counting. In fact, we match an analogous result of Fomin et al. [JACM 2016] for decision. This immediately yields, for a number of fundamental problems, parameterized approximate counting algorithms with the same running times as what is obtained for the decision variant using the representative family technique of Fomin et al. [JACM 2016]. For example, we devise an algorithm for approximately counting (a factor (1 ± ε) approximation algorithm) k-paths in an n-vertex directed graph (#k-Path) running in time O((2.619k + no(1)) · ε 12 · (n+m)). This improves over an earlier algorithm of Brand et al. [STOC 2018] that runs in time O(4k·kO(1)· ε 12 ·(n+m)). Additionally, we obtain an approximate counting analogue of the efficient computation of representative families for product families of Fomin et al. [TALG 2017], again essentially matching the running time for decision. This results in an algorithm with running time O((3.841k + |I|o(1)) · ε 16 · |I|) for computing a (1 + ε) approximation of the sum of the coefficients of the multilinear monomials in a degree-k homogeneous n-variate polynomial encoded by a monotone circuit (#Multilinear Monomial Detection). When restricted to monotone circuits (rather than polynomials of non-negative coefficients), this improves upon an earlier algorithm of Pratt [FOCS 2019] that runs in time 4.075k · ε 12 log 1 ε · nO(1)

Original languageEnglish
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2021
EditorsDaniel Marx
PublisherAssociation for Computing Machinery
Pages179-198
Number of pages20
ISBN (Electronic)9781611976465
StatePublished - 1 Jan 2021
Event32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States
Duration: 10 Jan 202113 Jan 2021

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Country/TerritoryUnited States
CityAlexandria, Virtual
Period10/01/2113/01/21

ASJC Scopus subject areas

  • Software
  • Mathematics (all)

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