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 $(1pm epsilon)/n_k$, where $n_k$ 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(maxleft{ left(frac{(nalpha)^{k/2}}{ n_k}right)^{frac{1}{k-1}} ,; minleft{nalpha,frac{nalpha^{k-1}}{n_k} 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 $n_3$ (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 $(1pm epsilon)$-approximation of $n_3$ 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 $n_k$ 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 $G$.
تحميل البحث