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 $O(\sqrt{n} \log (n/σ))$.

翻译：我们分析了未加权最小背包(MK)问题的平方和秩，即对于0/1变量在约束条件∑ᵢ₌₁ⁿ xᵢ ≥ q（q∈ℝ）下最小化∑ᵢ₌₁ⁿ xᵢ。此类实例长期作为理解布尔优化中提升-投影方法局限性的测试平台。例如，即使当q=1/2为常数时，Lovász-Schrijver和Sherali-Adams层级都需要（最大）秩n才能求解它们。当q=1/2时，SOS层级仅需次线性秩O(√n)即可求解未加权MK。另一方面，当q随n变化时，该问题的SOS秩可能变为线性。有趣的是，已知这种情况在q较大时以及q非常小时（0<q≤2⁻ⁿ）均会发生。这引发了一个问题：我们应将未加权MK的困难实例视为SOS层级的典型现象，还是阈值参数q的特定选择所致？本文通过展示参数q整个范围内未加权MK的SOS秩新上下界来回答此问题。当n-q≤O(1)时，我们证明SOS秩为常数。相比之下，当q≤O(1)且q指数级接近整数时，则需要线性秩。作为主要正面结果，我们表明对于q≤O(1)，线性秩极为罕见。这可用平滑分析的语言表达：在q添加均值为0、方差为σ²的高斯扰动后，MK的期望SOS秩为O(√n log(n/σ))。