We study the computational limits of the following general hypothesis testing problem. Let H=H_n be an \emph{arbitrary} undirected graph on n vertices. We study the detection task between a ``null'' Erd\H{o}s-R\'{e}nyi random graph G(n,p) and a ``planted'' random graph which is the union of G(n,p) together with a random copy of H=H_n. Our notion of planted model is a generalization of a plethora of recently studied models initiated with the study of the planted clique model (Jerrum 1992), which corresponds to the special case where H is a k-clique and p=1/2. Over the last decade, several papers have studied the power of low-degree polynomials for limited choices of H's in the above task. In this work, we adopt a unifying perspective and characterize the power of \emph{constant degree} polynomials for the detection task, when \emph{H=H_n is any arbitrary graph} and for \emph{any p=\Omega(1).} Perhaps surprisingly, we prove that the optimal constant degree polynomial is always given by simply \emph{counting stars} in the input random graph. As a direct corollary, we conclude that the class of constant-degree polynomials is only able to ``sense'' the degree distribution of the planted graph H, and no other graph theoretic property of it.

翻译：我们研究以下一般假设检验问题的计算极限。设H=H_n是n个顶点上的任意无向图。我们研究“零假设”下的Erdős-Rényi随机图G(n,p)与“植入”随机图（即G(n,p)与随机副本H=H_n的并集）之间的检测任务。我们的植入模型概念是对近期大量研究模型的推广，其源头可追溯至对植物团模型（Jerrum 1992）的研究，该模型对应于H为k-团且p=1/2的特殊情况。过去十年中，多篇论文研究了在有限选择H的上述任务中低次多项式的效力。本文采用统一视角，刻画了当H=H_n为任意图且p=Ω(1)时，常数次多项式在检测任务中的效力。令人惊讶的是，我们证明最优常数次多项式始终是通过简单统计输入随机图中的星星数量给出的。作为直接推论，我们得出结论：常数次多项式类仅能“感知”植入图H的度分布，而无法感知其任何其他图论性质。