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 omega >> log (n) added to an Erdos-Renyi graph G similar to G(n, 1/2), have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size omega = Omega(root n). By contrast, information theoretically, one can recover planted cliques so long as omega >> 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 omega >> root n/polylog n, improving on the bound omega >> n(1/3) 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].