We consider the problems from sublinear algorithms of sampling and counting edges from a graph on n vertices where our basic access is via uniformly sampled vertices. When we have accessed a vertex, we can see its degree, and we can access its neighbors, e.g., one picked uniformly at random. Accessing as few vertices as possible we want to sample and count edges. To appreciate our bounds below, note that if we have a graph with isolated vertices and a clique of size around √ m, then it takes Ω( n

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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.

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