Approximate Counting of k-Paths: Simpler, Deterministic, and in Polynomial Space

Daniel Lokshtanov, Andreas Björklund, Saket Saurabh, Meirav Zehavi

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

Recently, Brand et al. [STOC 2018] gave a randomized mathcal O(4k-2-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ϵ based on exterior algebra. Prior to our work, this has been the state-of-the-art. In this article, we revisit the algorithm by Alon and Gutner [IWPEC 2009, TALG 2010], and obtain the following results: •We present a deterministic 4k+ O(√k(log k+log2ϵ-1))m-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. •Additionally, we present a randomized 4k+mathcal O(logk(logk+logϵ-1))m-time polynomial-space algorithm. Our algorithm is simple - we only make elementary use of the probabilistic method. Here, n and m are the number of vertices and the number of edges, respectively. Additionally, our approach extends to approximate counting of other patterns of small size (such as q-dimensional p-matchings).

Original languageEnglish
Article number26
Pages (from-to)26:1-26:44
Number of pages44
JournalACM Transactions on Algorithms
Volume17
Issue number3
DOIs
StatePublished - 15 Jul 2021

Keywords

  • Parameterized complexity
  • k-path
  • parameterized counting problems

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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