Strong Low Degree Hardness for the Number Partitioning Problem

In the number partitioning problem (NPP) one aims to partition a given set of $N$ real numbers into two subsets with approximately equal sum. The NPP is a well-studied optimization problem and is famous for possessing a statistical-to-computational gap: when the $N$ numbers to be partitioned are i.i.d. standard gaussian, the optimal discrepancy is $2^{-\Theta(N)}$ with high probability, but the best known polynomial-time algorithms only find solutions with a discrepancy of $2^{-\Theta(\log^2 N)}$. This gap is a common feature in optimization problems over random combinatorial structures, and indicates the need for a study that goes beyond worst-case analysis. We provide evidence of a nearly tight algorithmic barrier for the number partitioning problem. Namely we consider the family of low coordinate degree algorithms (with randomized rounding into the Boolean cube), and show that degree $D$ algorithms fail to solve the NPP to accuracy beyond $2^{-\widetilde O(D)}$. According to the low degree heuristic, this suggests that simple brute-force search algorithms are nearly unimprovable, given any allotted runtime between polynomial and exponential in $N$. Our proof combines the isolation of solutions in the landscape with a conditional form of the overlap gap property: given a good solution to an NPP instance, slightly noising the NPP instance typically leaves no good solutions near the original one. In fact our analysis applies whenever the $N$ numbers to be partitioned are independent with uniformly bounded density.