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

TL;DR

This paper investigates the sum-of-squares hierarchy's rank complexity for unweighted minimum knapsack instances, revealing that high ranks are rare under typical parameter perturbations, with implications for understanding problem hardness.

Contribution

The authors provide new bounds on SOS rank for unweighted MK, showing linear rank is rare for most parameter regimes, especially after Gaussian perturbations.

Findings

SOS rank is constant when n-q is small.

Linear rank is needed when q is exponentially close to an integer.

Expected SOS rank after perturbation is O(√n) log(n/σ).

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 Lov\'asz-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 $\sigma^2$, the expected SOS rank of MK is $O(\sqrt{n} \log (n/\sigma))$.