On the integrality gap of degree-4 sum of squares for planted clique

The problem of finding large cliques in random graphs and its “planted” variant, where one wants to recover a clique of size ω log (n) added to an Erdős-Rényi graph G ∼ G(n,¹ ₂ ), have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size ω = Ω(n). By contrast, information theoretically, one can recover planted cliques so long as ω log (n). In this work, we continue the investigation of algorithms from the Sum of Squares hierarchy for solving the planted clique problem begun by Meka, Potechin, and Wigderson [2] and Deshpande and Montanari [25]. Our main result is that degree four SoS does not recover the planted clique unless ω n/ polylog n, improving on the bound ω n¹/³ due to Reference [25]. An argument of Kelner shows that the this result cannot be proved using the same certificate as prior works. Rather, our proof involves constructing and analyzing a new certificate that yields the nearly tight lower bound by “correcting” the certificate of References [2, 25, 27].