Fast Approximate Counting of Cycles

Keren Censor-Hillel, Tomer Even, Virginia Vassilevska Williams

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

Abstract

We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP’22] gave an algorithm that returns a (1 ± ε)-approximation in Õ(nω/tω−2) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n × n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1 ± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM (n, n/t1/(h−2), n)), the time to multiply n ×t1/(h−2) byt1/(h−2) × n matrices. n n Finally, we show that under popular fine-grained hypotheses, this running time is optimal.

Original languageEnglish
Title of host publication51st International Colloquium on Automata, Languages, and Programming, ICALP 2024
EditorsKarl Bringmann, Martin Grohe, Gabriele Puppis, Ola Svensson
ISBN (Electronic)9783959773225
DOIs
StatePublished - Jul 2024
Event51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 - Tallinn, Estonia
Duration: 8 Jul 202412 Jul 2024

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume297

Conference

Conference51st International Colloquium on Automata, Languages, and Programming, ICALP 2024
Country/TerritoryEstonia
CityTallinn
Period8/07/2412/07/24

Keywords

  • Approximate cycle counting Fast matrix multiplication
  • Approximate triangle counting
  • Fast rectangular matrix multiplication

All Science Journal Classification (ASJC) codes

  • Software

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