Finding and counting small tournaments in large tournaments

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Abstract

We present new algorithms for counting and detecting small tournaments in a given tournament. In particular, we prove that every tournament on four vertices (there are four) can be detected in O(n2) time and counted in O(nω) time where ω<2.372 is the matrix multiplication exponent. We further prove that any tournament on five vertices (there are 12) can be counted in O(nω+1) time. As for lower-bounds, we prove that for almost all k-vertex tournaments, the complexity of the detection problem is not easier than the complexity of the corresponding well-studied counting problem for undirected cliques of order k−O(log⁡k).

Original languageEnglish
Article number114911
JournalTheoretical Computer Science
Volume1024
DOIs
StatePublished - 12 Jan 2025

Bibliographical note

Publisher Copyright:
© 2024 Elsevier B.V.

Keywords

  • Counting
  • Detection
  • Tournament

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science

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