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arXiv · 2312.01419

Finding and counting small tournaments in large tournaments

Raphael Yuster
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Abstract

We present new algorithms for counting and detecting small tournaments in a given tournament. In particular, it is proved that every tournament on four vertices (there are four) can be detected in $O(n^2)$ time and counted in $O(n^\omega)$ time where $\omega < 2.373$ is the matrix multiplication exponent. It is also proved that any tournament on five vertices (there are $12$) can be counted in $O(n^{\omega+1})$ 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 {\em undirected cliques} of order $k-O(\log k)$.

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Raphael Yuster. 2023-12-03. Finding and counting small tournaments in large tournaments. https://arxiv.org/abs/2312.01419

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