Almost Optimal Bounds for Sublinear-Time Sampling of $k$-Cliques: Sampling Cliques is Harder Than Counting

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 $(1\pm \epsilon)/nk$, where $nk$ denotes the number of $k$-cliques in the graph and $\epsilon$ is a given approximation parameter. We prove that the query complexity of this problem is [ \Theta^{*\left(\max\left{} \left(\frac{(n\alpha)^{k/2}}{ nk}\right)^{{\frac{1}{k-1}}} ,\; \min\left{n\alpha,\frac{n\alpha^{k-1}}{nk} \right}\right}\right). ] where $n$ is the number of vertices in the graph, $\alpha$ is its arboricity, and $\Theta^*$ suppresses the dependence on $(\log n/\epsilon)^{O(k)}$. Interestingly, this establishes a separation between approximate counting and approximate uniform sampling in the sublinear regime. For example, if $k=3$, $\alpha = O(1)$, and $n3$ (the number of triangles) is $\Theta(n)$, then we get a lower bound of $\Omega(n^{1/4})$ (for constant $\epsilon$), while under these conditions, a $(1\pm \epsilon)$-approximation of $n3$ can be obtained by performing $\textrm{poly}(\log(n/\epsilon))$ queries (Eden, Ron and Seshadhri, SODA20). Our lower bound follows from a construction of a family of graphs with arboricity $\alpha$ such that in each graph there are $nk$ 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 $Hk$, 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 $Hk$, where the challenge is simulate queries to $H_k$ while being given access only to $G$.