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 $\Theta(\sqrt{n})$, polynomial-time algorithms for detection proliferate. We study faster -- namely, sublinear time -- algorithms in the high-signal regime when $k = \Theta(n^{1/2 + \delta})$, for some $\delta > 0$. 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 $\lvert M \rvert = \Theta(n^{\gamma})$, the clique can be detected when $\gamma > 3(1/2 - \delta)$ but not when $\gamma < 3(1/2 - \delta)$. As a result, the best known runtime for detecting a planted clique, $\widetilde{O}(n^{3(1/2-\delta)})$, 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.

翻译：我们考虑在 $n$ 个顶点的随机图中检测大小为 $k$ 的 planted clique 问题。当团的大小超过 $\Theta(\sqrt{n})$ 时，存在多种多项式时间检测算法。我们研究在信噪比高（即 $k = \Theta(n^{1/2 + \delta})$，其中 $\delta > 0$）的设定下更快速——即亚线性时间——的算法。为此，我们考虑一类非自适应查询算法：算法查询邻接矩阵中的某个子集 $M$ 的条目，然后计算所揭示条目的低度多项式函数。我们证明了此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 $\gamma > 3(1/2 - \delta)$ 时可检测到团，而当 $\gamma < 3(1/2 - \delta)$ 时则无法检测。由此可知，检测 planted clique 的最佳已知运行时间 $\widetilde{O}(n^{3(1/2-\delta)})$ 在非自适应低度类算法框架下无法进一步改进。我们基于条件低度似然比的下界证明揭示了 plantar clique 非自适应检测中的深层结构。通过将（有界化的）条件低度似然比作为势函数，我们证明：对任意非自适应查询模式，存在一个同样大小但结构高度规则的查询模式，其检测效果至少相当。