Abstract
There is a simple O(n³/{ε²T}) time algorithm for 1±εapproximate 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 Õ(n^ω). Is it possible to get a negative dependency on the number of triangles T while retaining the stateoftheart n^ω dependency on n? We answer this question positively by providing an algorithm which runs in time O({n^ω}/T^{ω2})⋅poly(n^o(1)/ε). 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. Our algorithm improves upon the state of the art when T ≫ 1 and T ≪ n.
We also consider the problem of approximate triangle counting in sparse graphs, parameterized by the number of edges m. The best known algorithm runs in time Õ_ε(m^{3/2}/T) [Eden et al., SIAM Journal on Computing, 2017]. An algorithm by Alon et al. [JACM, 1995] for exact triangle counting that runs in time Õ(m^{2ω/(ω + 1)}). We again get an algorithm whose complexity has a stateoftheart dependency on m while having negative dependency on T. Specifically, our algorithm runs in time O({m^{2ω/(ω+1)}}/{T^{2(ω1)/(ω+1)}}) ⋅ poly(n^o(1)/ε). This is again optimal in the sense that no better constant exponent of T is possible without worsening the dependency on m. This algorithm improves upon the state of the art when T ≫ 1 and T ≪ √m.
In both cases, algorithms with time complexity matching query complexity lower bounds were known on some range of parameters. While those algorithms have optimal query complexity for the whole range of T, the time complexity departs from the query complexity and is no longer optimal (as we show) for T ≪ n and T ≪ √m, respectively. We focus on the time complexity in this range of T. To the best of our knowledge, this is the first paper considering the discrepancy between query and time complexity in graph parameter estimation.
BibTeX  Entry
@InProceedings{tetek:LIPIcs.ICALP.2022.107,
author = {T\v{e}tek, Jakub},
title = {{Approximate Triangle Counting via Sampling and Fast Matrix Multiplication}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {107:1107:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772358},
ISSN = {18688969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16448},
URN = {urn:nbn:de:0030drops164485},
doi = {10.4230/LIPIcs.ICALP.2022.107},
annote = {Keywords: Approximate triangle counting, Fast matrix multiplication, Sampling}
}
Keywords: 

Approximate triangle counting, Fast matrix multiplication, Sampling 
Collection: 

49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) 
Issue Date: 

2022 
Date of publication: 

28.06.2022 