The arboricity captures the complexity of sampling edges

In this paper, we revisit the problem of sampling edges in an unknown graph G = (V, E) from a distribution that is (pointwise) almost uniform over E. We consider the case where there is some a priori upper bound on the arboriciy of G. Given query access to a graph G over n vertices and of average degree d and arboricity at most α, we design an algorithm that performs O ^α_d · ^log_ε^{3 n} queries in expectation and returns an edge in the graph such that every edge e ∈ E is sampled with probability (1 ± ε)/m. The algorithm performs two types of queries: degree queries and neighbor queries. We show that the upper bound is tight (up to poly-logarithmic factors and the dependence in ε), as Ω ^α_d queries are necessary for the easier task of sampling edges from any distribution over E that is close to uniform in total variational distance. We also prove that even if G is a tree (i.e., α = 1 so that ^α_d = Θ(1)), Ω _loglog^{log n}_n queries are necessary to sample an edge from any distribution that is pointwise close to uniform, thus establishing that a poly(log n) factor is necessary for constant α. Finally we show how our algorithm can be applied to obtain a new result on approximately counting subgraphs, based on the recent work of Assadi, Kapralov, and Khanna (ITCS, 2019).