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 $(1\pm ε)/n_k$, 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 \[ Θ^*\left(\max\left\{ \left(\frac{(nα)^{k/2}}{ n_k}\right)^{\frac{1}{k-1}} ,\; \min\left\{nα,\frac{nα^{k-1}}{n_k} \right\}\right\}\right). \] 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 $(1\pm ε)$-approximation of $n_3$ can be obtained by performing $\textrm{poly}(\log(n/ε))$ 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 $G$.