The Arboricity Captures the Complexity of Sampling Edges.

In this paper, we revisit the problem of sampling edges 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 $alpha$, we design an algorithm that performs $O!left(frac{alpha}{d} cdot frac{log^3 n}{varepsilon}right)$ queries expectation and returns an edge the graph such that every edge $e in E$ is sampled with probability $(1 pm varepsilon)/m$. 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 $varepsilon$), as $Omega!left(frac{alpha}{d} right)$ queries are necessary for the easier task of sampling edges from any distribution over $E$ that is close to uniform total variational distance. We also prove that even if $G$ is a tree (i.e., $alpha = 1$ so that $frac{alpha}{d}=Theta(1)$), $Omegaleft(frac{log n}{loglog n}right)$ queries are necessary to sample an edge from any distribution that is pointwise close to uniform, thus establishing that a $mathrm{poly}(log n)$ factor is necessary for constant $alpha$. 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).