We consider the random number partitioning problem (\texttt{NPP}): given a list $X\sim \mathcal{N}(0,I_n)$ of numbers, find a partition $\sigma\in\{-1,1\}^n$ with a small objective value $H(\sigma)=\frac{1}{\sqrt{n}}\left|\langle \sigma,X\rangle\right|$. The \texttt{NPP} is widely studied in computer science; it is also closely related to the design of randomized controlled trials. In this paper, we propose a planted version of the \texttt{NPP}: fix a $\sigma^*$ and generate $X\sim \mathcal{N}(0,I_n)$ conditional on $H(\sigma^*)\le 3^{-n}$. The \texttt{NPP} and its planted counterpart are statistically distinguishable as the smallest objective value under the former is $\Theta(\sqrt{n}2^{-n})$ w.h.p. Our first focus is on the values of $H(\sigma)$. We show that, perhaps surprisingly, planting does not induce partitions with an objective value substantially smaller than $2^{-n}$: $\min_{\sigma \ne \pm \sigma^*}H(\sigma) = \widetilde{\Theta}(2^{-n})$ w.h.p. Furthermore, we completely characterize the smallest $H(\sigma)$ achieved at any fixed distance from $\sigma^*$. Our second focus is on the algorithmic problem of efficiently finding a partition $\sigma$, not necessarily equal to $\pm\sigma^*$, with a small $H(\sigma)$. We show that planted \texttt{NPP} exhibits an intricate geometrical property known as the multi Overlap Gap Property ($m$-OGP) for values $2^{-\Theta(n)}$. We then leverage the $m$-OGP to show that stable algorithms satisfying a certain anti-concentration property fail to find a $\sigma$ with $H(\sigma)=2^{-\Theta(n)}$. Our results are the first instance of the $m$-OGP being established and leveraged to rule out stable algorithms for a planted model. More importantly, they show that the $m$-OGP framework can also apply to planted models, if the algorithmic goal is to return a solution with a small objective value.

翻译：我们考虑随机数划分问题（\texttt{NPP}）：给定一个列表$X\sim \mathcal{N}(0,I_n)$，寻找一个划分$\sigma\in\{-1,1\}^n$，使得目标函数$H(\sigma)=\frac{1}{\sqrt{n}}\left|\langle \sigma,X\rangle\right|$取值较小。\texttt{NPP}在计算机科学中被广泛研究，同时也与随机对照试验的设计密切相关。本文提出了\texttt{NPP}的植入版本：固定一个$\sigma^*$，在条件$H(\sigma^*)\le 3^{-n}$下生成$X\sim \mathcal{N}(0,I_n)$。由于原始问题中最小的目标函数值以高概率为$\Theta(\sqrt{n}2^{-n})$，\texttt{NPP}及其植入版本在统计上是可区分的。我们首先关注$H(\sigma)$的取值。令人惊讶的是，我们发现植入并不会导致与$\pm\sigma^*$不同的划分获得远小于$2^{-n}$的目标函数值：以高概率有$\min_{\sigma \ne \pm \sigma^*}H(\sigma) = \widetilde{\Theta}(2^{-n})$。此外，我们完整刻画了在距$\sigma^*$任意固定距离处所能达到的最小$H(\sigma)$值。其次，我们聚焦于高效寻找具有较小$H(\sigma)$的划分$\sigma$（不一定是$\pm\sigma^*$）的算法问题。研究表明，植入版\texttt{NPP}在值域$2^{-\Theta(n)}$上展现出一种称为多重重叠间隙性质（$m$-OGP）的精细几何结构。我们利用这一$m$-OGP性质证明，满足特定反集中性质的稳定算法无法找到$H(\sigma)=2^{-\Theta(n)}$的划分$\sigma$。这是首次在植入模型中建立并利用$m$-OGP性质来排除稳定算法。更重要的是，这表明当算法目标是返回具有较小目标函数值的解时，$m$-OGP框架同样适用于植入模型。