Planted Dense Subgraph (PDS) problem is a prototypical problem with a computational-statistical gap. It also exhibits an intriguing additional phenomenon: different tasks, such as detection or recovery, appear to have different computational limits. A detection-recovery gap for PDS was substantiated in the form of a precise conjecture given by Chen and Xu (2014) (based on the parameter values for which a convexified MLE succeeds) and then shown to hold for low-degree polynomial algorithms by Schramm and Wein (2022) and for MCMC algorithms for Ben Arous et al. (2020). In this paper, we demonstrate that a slight variation of the Planted Clique Hypothesis with secret leakage (introduced in Brennan and Bresler (2020)), implies a detection-recovery gap for PDS. In the same vein, we also obtain a sharp lower bound for refutation, yielding a detection-refutation gap. Our methods build on the framework of Brennan and Bresler (2020) to construct average-case reductions mapping secret leakage Planted Clique to appropriate target problems.

翻译：植入稠密子图问题是一个典型的具有计算-统计间隙的原型问题。该问题还表现出另一个引人注目的现象：不同任务（如检测或恢复）似乎具有不同的计算极限。针对植入稠密子图问题的检测-恢复间隙，Chen和Xu（2014）基于凸化极大似然估计成功的参数值提出了精确猜想，随后Schramm和Wein（2022）证明该猜想对低度多项式算法成立，Ben Arous等人（2020）证明其对马尔可夫链蒙特卡洛算法成立。本文证明，具有秘密泄露的植入团假设（由Brennan和Bresler（2020）提出）的轻微变体可推导出植入稠密子图问题的检测-恢复间隙。类似地，我们还获得了反驳的下界，进而得到检测-反驳间隙。我们的方法基于Brennan和Bresler（2020）的框架，构建了将秘密泄露植入团问题映射到相应目标问题的平均情形归约。