Almost Optimal Bounds for Sublinear-Time Sampling of k-Cliques: Sampling Cliques is Harder Than Counting
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In this work, we consider the problem of sampling a k-clique in a graph from an almost uniform distribution in sublinear time in the general graph query model. Specifically the algorithm should output each k-clique with probability (1±ϵ)/n_k, where n_k denotes the number of k-cliques in the graph and ϵ is a given approximation parameter. We prove that the query complexity of this problem is Θ^*(max{((nα)^k/2/ n_k)^1/k-1 , min{nα,nα^k-1/n_k}}). where n is the number of vertices in the graph, α is its arboricity, and Θ^* suppresses the dependence on (log n/ϵ)^O(k). Interestingly, this establishes a separation between approximate counting and approximate uniform sampling in the sublinear regime. For example, if k=3, α = O(1), and n_3 (the number of triangles) is Θ(n), then we get a lower bound of Ω(n^1/4) (for constant ϵ), while under these conditions, a (1±ϵ)-approximation of n_3 can be obtained by performing poly(log(n/ϵ)) queries (Eden, Ron and Seshadhri, SODA20). Our lower bound follows from a construction of a family of graphs with arboricity α such that in each graph there are n_k cliques (of size k), where one of these cliques is "hidden" and hence hard to sample. Our upper bound is based on defining a special auxiliary graph H_k, such that sampling edges almost uniformly in H_k 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 H_k, where the challenge is simulate queries to H_k while being given access only to G.
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