The Planted Number Partitioning Problem

Given a list $X\sim \mathcal{N}(0,I_n)$ of numbers, the random number partitioning problem (NPP) seeks a partition $\boldsymbol{\sigma}\in$ {-1,1}$^n$ with a small $H(\boldsymbol{\sigma})=\frac{1}{\sqrt{n}}\left|⟨\boldsymbol{\sigma},X⟩\right|$. The NPP has been extensively studied in computer science, probability and combinatorics; it is also closely linked to covariate balancing and randomized controlled trials. We introduce a planted version of the random NPP: fix a $\boldsymbol{\sigma}$* and generate $X\sim \mathcal{N}(0,I_n)$ conditional on $H(\boldsymbol{\sigma^*})\le 3^{-n}$. The random and planted models are statistically distinguishable, since in the former case $\min_{\boldsymbol{\sigma}}H(\boldsymbol{\sigma})=\Theta(\sqrt{n}2^{-n})$ w.h.p. We first analyze the values of $H(\boldsymbol{\sigma})$. We show that, perhaps surprisingly, planting does not yield partitions with objective values substantially smaller than $2^{-n}$: we have $\min_{\boldsymbol{\sigma} \ne \pm \boldsymbol{\sigma}}$* $H(\boldsymbol{\sigma}) = \widetilde{\Theta}(2^{-n})$ w.h.p. Moreover, we precisely characterize the minimal $H(\boldsymbol{\sigma})$ achievable at any fixed distance from $\boldsymbol{\sigma^*}$. Turning to the algorithmic problem, we ask whether one can efficiently find a partition $\boldsymbol{\sigma}$ with small $H(\boldsymbol{\sigma})$. We prove that planted NPP exhibits the multi Overlap Gap Property ($m$-OGP) at scales $2^{-\Theta(n)}$. Building on this barrier, we show that stable algorithms satisfying a natural anti-concentration property cannot find partitions with $H(\boldsymbol{\sigma})=2^{-\Theta(n)}$. This is the first instance where the $m$-OGP rules out stable algorithms in a planted setting. Our results demonstrate that the multi OGP framework, previously developed for unplanted models, extends naturally to planted ones when the goal is to recover low-objective solutions. They further point to a statistical–computational gap: although the random and planted NPP are statistically distinguishable, we conjecture that no polynomial-time algorithm can distinguish them with nontrivial advantage. Our results demonstrate that planted NPP harbors intriguing features and it is a particularly promising model for probing algorithmic barriers in planted problems.