Planted Random Number Partitioning Problem

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.