改进的检测与恢复植入 $\Theta(\sqrt{n})$-团的多项式时间算法

中文摘要 AI 辅助

在植入团问题中，观察者要么看到一个在 $n$ 个顶点上的 Erdős–Rényi 图，要么看到这样的图加上一个添加到 $k = k(n)$ 个顶点上的团，并寻求检测或恢复该团。人们普遍认为 $k = \Theta(\sqrt{n})$ 是存在多项式时间算法完成这些任务的最小团大小。我们在此机制下开发了新算法，使用颜色编码来估计带符号子图计数，并通过快速矩阵乘法进一步加速。我们首先证明，对于每个 $t \geq 1$，其中 $c(t)$ 是与树宽至多 $t$ 的连通图数量增长阶相关的常数，大小为 $k = \lambda\sqrt{n}$ 且随机位置植入的团，当 $\lambda > 1 / \sqrt{c(t)}$ 时，可以在 $n^{t + 1 + o(1)}$ 时间内被检测和恢复。例如，由于 $c(1) = e$，这通过计数带符号树恢复了 Deshpande–Montanari (2015) 的 $\widetilde{O}(n^2)$ 时间消息传递算法的性能，该算法在 $\lambda > 1 / \sqrt{e} \approx 0.6066$ 时成功。对于 $t \geq 3$，$c(t)$ 的精确值未知，但对其下界给出了一个较慢多项式时间算法的层次结构，这些算法在更小的 $\lambda$ 下成功。我们进一步证明，上述 $t = 2$ 的算法可以在 $n^{\omega + o(1)}$ 时间内实现，其中 $\omega$ 是方阵乘法的常数，并在 $\lambda > 0.3320$ 时成功；在 $\omega = 2$ 的民间猜想下，该算法以 Deshpande–Montanari 算法的近线性时间运行，同时找到更小的团。其次，我们证明上述 $t = 1$ 的算法可以与 Alon–Krivelevich–Sudakov (1998) 的增强方案结合，使用矩形矩阵乘法，为更小的 $\lambda$ 提供改进的运行时间。综合来看，我们的结果实现了运行时间与信号强度 $\lambda$ 之间已知的最佳权衡。

英文摘要

In the planted clique problem, one observes either an Erdős--Rényi graph on $n$ vertices or such a graph with a clique added to $k = k(n)$ vertices, and seeks to detect or recover the clique. It is widely believed that $k = Θ(\sqrt{n})$ is the smallest clique size for which polynomial-time algorithms exist for these tasks. We develop new algorithms in this regime using color-coding to estimate signed subgraph counts, further accelerated with fast matrix multiplication. We first show that, for each $t \geq 1$, for $c(t)$ a constant associated to the order of growth of the number of connected graphs of treewidth at most $t$, cliques of size $k = λ\sqrt{n}$ planted in a random location with $λ> 1 / \sqrt{c(t)}$ can be detected and recovered in time $n^{t + 1 + o(1)}$. For instance, since $c(1) = e$, this recovers by counting signed trees the performance of the $\widetilde{O}(n^2)$-time message-passing algorithm of Deshpande--Montanari (2015) that succeeds when $λ> 1 / \sqrt{e} \approx 0.6066$. For $t \geq 3$, the exact value of $c(t)$ is not known, but lower bounds on it give a hierarchy of slower polynomial-time algorithms that succeed for smaller $λ$. We further show that the above algorithm for $t = 2$ can be implemented in time $n^{ω+ o(1)}$ for $ω$ the constant of square matrix multiplication and succeeds when $λ> 0.3320$; under the folklore conjecture that $ω= 2$, this runs in the nearly-linear time of the algorithm of Deshpande--Montanari while finding smaller cliques. Second, we show that the above algorithm for $t = 1$ can be combined with the boosting scheme of Alon--Krivelevich--Sudakov (1998) using rectangular matrix multiplication, giving improved runtimes for smaller $λ$. Taken together, our results achieve the best known tradeoff between runtime and signal strength $λ$.

发表机构

• Johns Hopkins University（约翰斯·霍普金斯大学）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

77 pages