Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time.

We present a simple sublinear-time algorithm for sampling an arbitrary subgraph $H$ \emph{exactly uniformly} from a graph $G$, to which the algorithm has access by performing the following types of queries: (1) uniform vertex queries, (2) degree queries, (3) neighbor queries, (4) pair queries and (5) edge sampling queries. The query complexity and running time of our algorithm are $\tilde{O}(\min\{m, \frac{m^{\rho(H)}}{\# H}\})$ and $\tilde{O}(\frac{m^{\rho(H)}}{\# H})$, respectively, where $\rho(H)$ is the fractional edge-cover of $H$ and $\# H$ is the number of copies of $H$ in $G$. For any clique on $r$ vertices, i.e., $H=K_r$, our algorithm is almost optimal as any algorithm that samples an $H$ from any distribution that has $\Omega(1)$ total probability mass on the set of all copies of $H$ must perform $\Omega(\min\{m, \frac{m^{\rho(H)}}{\# H\cdot (cr)^r}\})$ queries. Together with the query and time complexities of the $(1\pm \varepsilon)$-approximation algorithm for the number of subgraphs $H$ by Assadi, Kapralov and Khanna [ITCS 2018] and the lower bound by Eden and Rosenbaum [APPROX 2018] for approximately counting cliques, our results suggest that in our query model, approximately counting cliques is ``equivalent to'' exactly uniformly sampling cliques, in the sense that the query and time complexities of exactly uniform sampling and randomized approximate counting are within a polylogarithmic factor of each other. This stands in interesting contrast to an analogous relation between approximate counting and almost uniformly sampling for self-reducible problems in the polynomial-time regime by Jerrum, Valiant and Vazirani [TCS 1986].