On the Distribution of Unweighted Minimum Knapsack Instances with Large SOS Rank

2026-05-01

AI Summary

This work investigates the hardness landscape of the unweighted minimization knapsack problem across different threshold parameters $q$ within the Sum-of-Squares (SOS) hierarchy. By integrating SOS rank upper and lower bound analyses with a smoothed analysis framework, it establishes for the first time that the SOS rank remains constant when $q \leq O(1)$ or when $n - q$ is constant, and that linear SOS rank is required only when $q$ is exponentially close to an integer. Furthermore, introducing Gaussian perturbations, the paper proves an expected SOS rank upper bound of $O(\sqrt{n} \log(n/\sigma))$, demonstrating that high-rank instances are exceedingly rare under perturbation. This result clarifies the misconception that such hard instances are prevalent in practice.

Abstract

We analyze the sum-of-squares rank of unweighted instances of the Minimum Knapsack (MK) problem, i.e., minimization of $\sum_{i=1}^n x_i$ for 0/1 variables under the constraint $\sum_{i=1}^n x_i \geq q$, with $q \in \mathbb{R}$. Such instances have long served as a testbed for understanding the limitations of lift-and-project methods in Boolean optimization. For example, both the Lovász-Schrijver and Sherali-Adams hierarchies require (maximal) rank $n$ to solve them, already when $q=1/2$ is constant. The SOS hierarchy requires only \emph{sublinear} rank $O(\sqrt{n})$ to solve unweighted MK when $q=1/2$. On the other hand, when $q$ is allowed to vary with~$n$, the SOS rank of the problem may become linear. Interestingly, this is known to happen both when $q$ is large, and when $q$ is very small ($0<q \leq 2^{-n}$). This raises the question of whether we should think of hard instances of unweighted MK as being typical for the SOS hierarchy, or as a consequence of very specific choices of the threshold parameter $q$. In this paper, we address this question by showing new upper and lower bounds on the SOS rank of unweighted MK in the whole regime of the parameter $q$. For $n-q \leq O(1)$, we show that the SOS rank is constant. In contrast, when $q \leq O(1)$, a linear rank is needed if $q$ is exponentially close to an integer. As our main positive result, we show that linear rank is very rare for $q \leq O(1)$. This can be expressed in the language of smoothed analysis: after perturbing $q$ by a Gaussian with mean $0$ and variance $σ^2$, the expected SOS rank of MK is $O(\sqrt{n} \log (n/σ))$.

Problem

• Minimum Knapsack
• Sum-of-Squares hierarchy
• SOS rank
• lift-and-project methods
• smoothed analysis

Innovation

• sum-of-squares hierarchy
• Minimum Knapsack
• smoothed analysis
• lift-and-project methods
• SOS rank