We prove that for every $D \in \N$, and large enough constant $d \in \N$, with high probability over the choice of $G \sim G(n,d/n)$, the \Erdos-\Renyi random graph distribution, the canonical degree $2D$ Sum-of-Squares relaxation fails to certify that the largest independent set in $G$ is of size $o(\frac{n}{\sqrt{d} D^4})$. In particular, degree $D$ sum-of-squares strengthening can reduce the integrality gap of the classical \Lovasz theta SDP relaxation by at most a $O(D^4)$ factor. This is the first lower bound for $>4$-degree Sum-of-Squares (SoS) relaxation for any problems on \emph{ultra sparse} random graphs (i.e. average degree of an absolute constant). Such ultra-sparse graphs were a known barrier for previous methods and explicitly identified as a major open direction (e.g.,~\cite{deshpande2019threshold, kothari2021stressfree}). Indeed, the only other example of an SoS lower bound on ultra-sparse random graphs was a degree-4 lower bound for Max-Cut. Our main technical result is a new method to obtain spectral norm estimates on graph matrices (a class of low-degree matrix-valued polynomials in $G(n,d/n)$) that are accurate to within an absolute constant factor. All prior works lose $\poly log n$ factors that trivialize any lower bound on $o(\log n)$-degree random graphs. We combine these new bounds with several upgrades on the machinery for analyzing lower-bound witnesses constructed by pseudo-calibration so that our analysis does not lose any $\omega(1)$-factors that would trivialize our results. In addition to other SoS lower bounds, we believe that our methods for establishing spectral norm estimates on graph matrices will be useful in the analyses of numerical algorithms on average-case inputs.

翻译：我们证明，对于任意 $D \in \N$ 及足够大的常数 $d \in \N$，以高概率从 \Erdos-\Renyi 随机图分布 $G \sim G(n,d/n)$ 中采样得到图 $G$ 时，经典的 $2D$ 次平方和规划松弛法无法证明 $G$ 中最大独立集的规模为 $o(\frac{n}{\sqrt{d} D^4})$。特别地，$D$ 次平方和规划强化最多只能将经典 \Lovasz theta 半定规划松弛的整数性间隙缩小 $O(D^4)$ 倍。这是针对\emph{超稀疏}随机图（即平均度为绝对常数的随机图）上任意问题的 $>4$ 次平方和规划松弛的首个下界结果。此类超稀疏图曾是先前方法的已知障碍，并被明确列为重要开放方向（例如~\cite{deshpande2019threshold, kothari2021stressfree}）。实际上，超稀疏随机图上仅有的另一个平方和规划下界结果是 Max-Cut 问题的 4 次下界。我们的核心技术成果是提出了一种新方法，用于获得图矩阵（$G(n,d/n)$ 中的一类低次矩阵值多项式）的谱范数估计，其精度可达绝对常数因子级别。所有先前工作均存在 $\poly log n$ 因子的损失，这使得任何针对 $o(\log n)$ 次随机图的下界结果失效。我们将这些新界与伪校准构造的下界见证分析机制的多项改进相结合，从而避免了任何会导致结果失效的 $\omega(1)$ 因子损失。除其他平方和规划下界外，我们相信所提出的图矩阵谱范数估计方法将有助于分析平均情况输入下数值算法的性能。