Constant Approximation of Arboricity in Near-Optimal Sublinear Time

Abstract: We present a randomized algorithm that computes a constant approximation of a graph's arboricity, using $\tilde{O}(n/λ)$ queries to adjacency lists and in the same time bound. Here, $n$ and $λ$ denote the number of nodes and the graph's arboricity, respectively. The $\tilde{O}(n/λ)$ query complexity of our algorithm is nearly optimal. Our constant approximation settles a question of Eden, Mossel, and Ron [SODA'22], who achieved an $O(\log² n)$ approximation with the same query and time complexity and asked whether a better approximation can be achieved using near-optimal query complexity. A key technical challenge in the problem is due to recursive algorithms based on probabilistic samplings, each with a non-negligible error probability. In our case, many of the recursions invoked could have bad probabilistic samples and result in high query complexities. The particular difficulty is that those bad recursions are not easy or cheap to detect and discard. Our approach runs multiple recursions in parallel, to attenuate the error probability, using a careful \textit{scheduling mechanism} that manages the speed at which each of them progresses and makes our overall query complexity competitive with the single good recursion. We find this usage of parallelism and scheduling in a sublinear algorithm remarkable, and we are hopeful that similar ideas may find applications in a wider range of sublinear algorithms that rely on probabilistic recursions.