There is a trivial algorithm for approximatetriangle counting where $T$ is the number of triangles in the graph and $n$ thenumber of vertices . At the same time, one may count triangles exactly usingfast matrix multiplication in time $O . We answer this question positively by providing an algorithm which runs in time . This is optimal in the sense that as long as the exponent of$T is independent of $n, T$, it cannot be improvedwhile retaining the dependency on $n . We also consider the problem of approximate triangle counting in sparsegraphs, parameterizing by number of edges$m$. The best known algorithmruns in time$\tilde{O\big(\frac{m^{3/2)$[Eden et al., SIAM Journalon Computing, 2017]. There is also also a well known algorithm for exact trianglecounting that runs in$tribal counting that run in time. We again get an algorithm that runs . This algorithm that retains the exponent \$m

Author(s) : Jakub Tětek