Almost Optimal Bounds for Sublinear-Time Sampling of k-Cliques in Bounded Arboricity Graphs

Counting and sampling small subgraphs are fundamental algorithmic tasks. Motivated by the need to handle massive datasets efficiently, recent theoretical work has examined the problems in the sublinear time regime. In this work, we consider the problem of sampling a k-clique in a graph from an almost uniform distribution. Specifically the algorithm should output each k-clique with probability (1 ± ϵ)/nk, where nk denotes the number of k-cliques in the graph and ϵ is a given approximation parameter. To this end, the algorithm may perform degree, neighbor, and pair queries. We focus on the class of graphs with arboricity at most α, and prove that the query complexity of the problem is (Equation presented) where n is the number of vertices in the graph, and Θ^∗(·) suppresses dependencies on (log n/ϵ)^O(k). Our upper bound is based on defining a special auxiliary graph Hk, such that sampling edges almost uniformly in Hk translates to sampling k-cliques almost uniformly in the original graph G. We then build on a known edge-sampling algorithm (Eden, Ron and Rosenbaum, ICALP19) to sample edges in Hk. The challenge is simulating queries to Hk while being given query access only to G. Our lower bound follows from a construction of a family of graphs with arboricity α such that in each graph there are nk k-cliques, where one of these cliques is “hidden” and hence hard to sample.