Fast Approximate Counting of Cycles
<|reference_start|>Fast Approximate Counting of Cycles: We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, T\v{e}tek [ICALP'22] gave an algorithm that returns a $(1 \pm \eps)$-approximation in $\tilde{O}(n^\omega/t^{\omega-2})$ time, where $t$ is the unknown number of triangles in the given $n$ node graph and $\omega<2.372$ is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an $n\times n/t$ matrix by an $n/t \times n$ matrix. We then extend our framework to obtain the first nontrivial $(1 \pm \eps)$-approximation algorithms for the number of $h$-cycles in a graph, for any constant $h\geq 3$. Our running time is   \[\tilde{O}(\mathsf{MM}(n,n/t^{1/(h-2)},n)), \textrm{the time to multiply } n\times \frac{n}{t^{1/(h-2)}} \textrm{ by } \frac{n}{t^{1/(h-2)}}\times n \textrm{ matrices}.\]   Finally, we show that under popular fine-grained hypotheses, this running time is optimal.<|reference_end|>
@article{censor-hillel2024fast,
    title={Fast Approximate Counting of Cycles},
    author={Keren Censor-Hillel, Tomer Even and Virginia Vassilevska Williams},
    journal={arXiv preprint arXiv:2409.19292},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.19292},
    primaryClass={cs.DS}
}