Sum-of-Squares Lower Bounds for Independent Set in Ultra-Sparse Random Graphs

TL;DR

This paper establishes new lower bounds for the Sum-of-Squares hierarchy in certifying independent set sizes in ultra-sparse random graphs, revealing limitations of current relaxations and introducing novel spectral norm estimation techniques.

Contribution

It provides the first non-trivial lower bounds for degree >4 Sum-of-Squares relaxations on ultra-sparse graphs and develops new spectral norm estimation methods for graph matrices.

Findings

Degree 2D SoS fails to certify large independent sets in ultra-sparse graphs.

Degree D SoS can only improve the Lovász theta relaxation by a factor of O(D^4).

New spectral norm bounds on graph matrices that avoid polylogarithmic losses.

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 $\omega(1)$-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.