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arXiv 2402.05451cs.DScs.CCstat.ML
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亚线性时间内检测 planted clique 的低度相变

Low-degree phase transitions for detecting a planted clique in sublinear time

Jay Mardia, Kabir Aladin Verchand, Alexander S. Wein

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23 pages, 2 figures
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AI总结：

本文研究亚线性时间内检测 planted clique 的问题，证明非自适应低度算法存在计算相变，并揭示最优查询模式的结构性质。

摘要：

我们考虑在 $n$ 个顶点的随机图中检测大小为 $k$ 的 planted clique 的问题。当团的大小超过 $\Theta(\sqrt{n})$ 时，多项式时间检测算法大量涌现。我们研究在高信号区域（即 $k = \Theta(n^{1/2 + \delta})$，其中 $\delta > 0$）中更快的算法——即亚线性时间算法。为此，我们考虑非自适应地查询邻接矩阵的一个子集 $M$ 的条目，然后计算所揭示条目的低度多项式函数的算法。我们证明了这类非自适应低度算法的计算相变：在缩放 $\lvert M \rvert = \Theta(n^\gamma)$ 下，当 $\gamma > 3(1/2 - \delta)$ 时可以检测到团，但当 $\gamma < 3(1/2 - \delta)$ 时则不能。因此，已知最佳的检测 planted clique 的运行时间 $\widetilde{O}(n^{3(1/2-\delta)})$ 无法在不超越非自适应低度类的情况下得到改进。我们的下界证明——基于对条件低度似然比的界定——揭示了非自适应检测 planted clique 的进一步结构。使用（对）条件低度似然比（的界）作为势函数，我们证明对于每个非自适应查询模式，存在一个相同大小的高度结构化查询模式，其效果至少相同。

英文摘要：

We consider the problem of detecting a planted clique of size $k$ in a random graph on $n$ vertices. When the size of the clique exceeds $Θ(\sqrt{n})$, polynomial-time algorithms for detection proliferate. We study faster -- namely, sublinear time -- algorithms in the high-signal regime when $k = Θ(n^{1/2 + δ})$, for some $δ> 0$. To this end, we consider algorithms that non-adaptively query a subset $M$ of entries of the adjacency matrix and then compute a low-degree polynomial function of the revealed entries. We prove a computational phase transition for this class of non-adaptive low-degree algorithms: under the scaling $\lvert M \rvert = Θ(n^γ)$, the clique can be detected when $γ> 3(1/2 - δ)$ but not when $γ< 3(1/2 - δ)$. As a result, the best known runtime for detecting a planted clique, $\widetilde{O}(n^{3(1/2-δ)})$, cannot be improved without looking beyond the non-adaptive low-degree class. Our proof of the lower bound -- based on bounding the conditional low-degree likelihood ratio -- reveals further structure in non-adaptive detection of a planted clique. Using (a bound on) the conditional low-degree likelihood ratio as a potential function, we show that for every non-adaptive query pattern, there is a highly structured query pattern of the same size that is at least as effective.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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