We aim to understand the extent to which the noise distribution in a planted signal-plus-noise problem impacts its computational complexity. To that end, we consider the planted clique and planted dense subgraph problems, but in a different ambient graph. Instead of Erd\H{o}s-R\'enyi $G(n,p)$, which has independent edges, we take the ambient graph to be the \emph{random graph with triangles} (RGT) obtained by adding triangles to $G(n,p)$. We show that the RGT can be efficiently mapped to the corresponding $G(n,p)$, and moreover, that the planted clique (or dense subgraph) is approximately preserved under this mapping. This constitutes the first average-case reduction transforming dependent noise to independent noise. Together with the easier direction of mapping the ambient graph from Erd\H{o}s-R\'enyi to RGT, our results yield a strong equivalence between models. In order to prove our results, we develop a new general framework for reasoning about the validity of average-case reductions based on \emph{low sensitivity to perturbations}.

翻译：我们旨在理解种植信号加噪声问题中噪声分布对其计算复杂性的影响程度。为此，我们考虑种植团和种植稠密子图问题，但采用不同的背景图。与具有独立边的Erdős-Rényi $G(n,p)$不同，我们选取的背景图是通过在$G(n,p)$中添加三角形而获得的\emph{带三角随机图}（RGT）。我们证明RGT可高效映射至对应的$G(n,p)$，且在此映射下种植团（或稠密子图）近似保持不变。这构成了首个将依赖噪声转化为独立噪声的平均情形归约。结合将背景图从Erdős-Rényi映射到RGT的较易方向，我们的结果揭示了模型间的强等价性。为证明结论，我们发展了一个基于\emph{对扰动的低敏感性}的新通用框架，用于论证平均情形归约的有效性。