Sum-of-Squares Lower Bounds for Independent Set in Ultra-Sparse Random Graphs
Pravesh Kothari, Aaron Potechin, Jeff Xu
TL;DR
This work establishes the first strong Sum-of-Squares lower bound for independent set on ultra-sparse random graphs, showing that degree-$2D$ SoS cannot certify bounds better than $o\left(\frac{n}{\sqrt{d}\,D^4}\right)$ for $G\sim G(n,d/n)$, thereby bounding improvements over the Lovász theta SDP by at most a factor of $O(D^4)$. The authors develop a sharp, local spectral-norm analysis of graph matrices built from sparse graphs, introducing a localization technique for trace moments and an upgraded pseudo-calibration machinery that avoids polylogarithmic losses. A key technical achievement is establishing absolute-constant spectral-norm bounds for line-graph and Z-shape graph matrices on truncated ultra-sparse graphs, enabling precise PSDness arguments for the SoS moment matrix. The approach hinges on a fine-grained block-value framework that ties vertex-separator structure to norm bounds, and it is complemented by a careful treatment of pruning, 2-cycle freeness, and singleton edges. The resulting framework not only yields the main lower bound but also provides tools potentially useful for analyzing numerical algorithms on average-case inputs in sparse regimes.
Abstract
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 $ω(1)$-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.