Fast Approximate Counting of Cycles

TL;DR

This paper presents improved algorithms for approximately counting cycles of fixed length in directed graphs, achieving near-optimal running times based on matrix multiplication, and extends results from triangles to longer cycles.

Contribution

The authors develop faster approximate counting algorithms for cycles of any fixed length in directed graphs, generalizing previous triangle-counting methods and establishing their near-optimality.

Findings

New algorithms match matrix multiplication time for cycle counting.

First nontrivial approximation algorithms for cycles of length h ≥ 3.

Proved the optimality of these algorithms under certain hypotheses.

Abstract

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.