Sum of Squares Lower Bounds for Sparse Independent Set
December 8, 2021

Abstract: The Sum-of-Squares (SoS) hierarchy of semidefinite programs is a powerful algorithmic paradigm which captures state-of-the-art algorithmic guarantees for a wide array of problems. In the average case setting, SoS lower bounds provide strong evidence of algorithmic hardness or information-computation gaps. Prior to this work, SoS lower bounds have been obtained for problems in the “dense” input regime, while the sparse regime has remained out of reach. We make the first progress in this direction by obtaining strong SoS lower bounds for the problem of Independent Set on sparse random graphs. We prove that with high probability over an Erdõs–Rényi random graph $G \sim G_{n,d/n}$ with average degree $d > (\log n)^2$, degree-D SoS fails to refute the existence of an independent set of size

$k = \Omega(\frac{n}{\sqrt{d}(\log n)(D)^{c_0}})$ in G (where $c_0$ is an absolute constant), whereas the true size of the largest independent set in G is $O(\frac{n \log d}{d})$.