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 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 low sensitivity to perturbations.

翻译：我们旨在理解植入信号+噪声问题中噪声分布对其计算复杂程度的影响。为此，我们考虑植入团和植入稠密子图问题，但采用不同的背景图。不同于具有独立边的Erdős-Rényi $G(n,p)$，我们以通过向$G(n,p)$添加三角形得到的含三角形随机图(RGT)作为背景图。我们证明RGT可被高效映射至对应的$G(n,p)$，并且植入团（或稠密子图）在该映射下近似保持不变。这构成了首个将依赖噪声转换为独立噪声的平均复杂度归约。结合从Erdős-Rényi到RGT的背景图映射的更易方向，我们的结果建立了模型间的强等价性。为证明这些结果，我们发展了一个基于低扰动敏感性的平均复杂度归约有效性推理新通用框架。