Approximate Triangle Counting via Sampling and Fast Matrix Multiplication

Abstract

There is a trivial Õ( n 3 T ) 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 Õ(n ω ). Is it possible to get a negative dependency on the number of triangles T while retaining the n ω dependency on n? We answer this question positively by providing an algorithm which runs in time O n ω T ω−2 • poly(n o(1) / ). 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, as follows from the lower bound of Eden and Rosenbaum [APPROX/RANDOM 2018].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 Õ m 3/2 T [Eden et al., SIAM Journal on Computing, 2017]. There is also a well known algorithm for exact triangle counting that runs in time Õ(m 2ω/(ω+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 O m 2ω/(ω+1)T 2(ω−1)/(ω+1) • poly(n o(1) / ). This is again optimal in the sense that if the exponent of T is to be constant, it cannot be improved.

Edge Sampling and Graph Parameter Estimation via Vertex Neighborhood Accesses

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