论文标题
在均方根时间完全均匀地对任意子图进行采样
Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time
论文作者
论文摘要
我们提出了一种简单的弹力时间算法，用于对带有$ m $边缘的图$ g $进行任意子段$ h $ \ emph {完全均匀的emph {agriarly comph {emph {agries，（（2）Queinaries，（2）Queries，（3）Queries和（4）Queries和（4）Queries和（4）Same Same Same Same-egries coolies：（1）度量查询：（1）度量查询：我们算法的查询复杂性和运行时间为$ \ tilde {o}（\ min \ {m，\ frac {m^{ρ（h）}} {\＃h} \} \} \} \}）$ and $ \ tilde {o}（O}（o}（o）{\ frac {\ frac {m^） $ρ（h）$是$ h $的分数边缘，$ \＃h $是$ g $中$ h $的副本数。对于$ r $顶点的任何集团，即$ h = k_r $，我们的算法几乎是任何算法，这些算法从任何具有$ω（1）$总概率的分布中示例$ h $的算法，$ h $ of $ h $ of $ h $ of $ h $均必须执行$ω（ 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从某种意义上说，统一的采样库是彼此彼此之间的均匀抽样和随机近似计数的查询和时间复杂性。这与近似计数和几乎均匀地对Jerrum，Valiant和Vazirani在多项式时间制度中的自我还原问题进行了类似关系形成鲜明对比[TCS 1986]。
We present a simple sublinear-time algorithm for sampling an arbitrary subgraph $H$ \emph{exactly uniformly} from a graph $G$ with $m$ edges, to which the algorithm has access by performing the following types of queries: (1) degree queries, (2) neighbor queries, (3) pair queries and (4) edge sampling queries. The query complexity and running time of our algorithm are $\tilde{O}(\min\{m, \frac{m^{ρ(H)}}{\# H}\})$ and $\tilde{O}(\frac{m^{ρ(H)}}{\# H})$, respectively, where $ρ(H)$ 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 $Ω(1)$ total probability mass on the set of all copies of $H$ must perform $Ω(\min\{m, \frac{m^{ρ(H)}}{\# H\cdot (cr)^r}\})$ 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].
