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- W4287208421 abstract "There is a trivial $O(frac{n^3}{T})$ time algorithm for approximate triangle counting where $T$ is the number of triangles in the graph and $n$ the number of vertices. At the same time, one may count triangles exactly using fast matrix multiplication in time $tilde{O}(n^omega)$. Is it possible to get a negative dependency on the number of triangles $T$ while retaining the $n^omega$ dependency on $n$? We answer this question positively by providing an algorithm which runs in time $Obig(frac{n^omega}{T^{omega - 2}}big) cdot text{poly}(n^{o(1)}/epsilon)$. This is optimal in the sense that as long as the exponent of $T$ is independent of $n, T$, it cannot be improved while retaining the dependency on $n$; this as follows from the lower bound of Eden and Rosenbaum [APPROX/RANDOM 2018]. Our algorithm improves upon the state of the art when $T = omega(1)$ and $T = o(n)$. We also consider the problem of approximate triangle counting in sparse graphs, parameterizing by the number of edges $m$. The best known algorithm runs in time $tilde{O}big(frac{m^{3/2}}{T}big)$ [Eden et al., SIAM Journal on Computing, 2017]. There is also a well known algorithm for exact triangle counting that runs in time $tilde{O}(m^{2omega/(omega + 1)})$. We again get an algorithm that retains the exponent of $m$ while running faster on graphs with larger number of triangles. Specifically, our algorithm runs in time $OBig(frac{m^{2omega/(omega+1)}}{ T^{2(omega-1)/(omega+1)}}Big) cdot text{poly}(n^{o(1)}/epsilon)$. This is again optimal in the sense that if the exponent of $T$ is to be constant, it cannot be improved without worsening the dependency on $m$. This algorithm improves upon the state of the art when $T = omega(1)$ and $T = o(sqrt{m})$." @default.
- W4287208421 created "2022-07-25" @default.
- W4287208421 creator A5077723390 @default.
- W4287208421 date "2021-04-17" @default.
- W4287208421 modified "2023-09-27" @default.
- W4287208421 title "Approximate Triangle Counting via Sampling and Fast Matrix Multiplication" @default.
- W4287208421 doi "https://doi.org/10.48550/arxiv.2104.08501" @default.
- W4287208421 hasPublicationYear "2021" @default.
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