Tight Lower Bounds for Planted Clique in the Degree-4 SOS Program

We give a lower bound of Ω̃( √ n) for the degree-4 Sum-of-Squares SDP relaxation for the planted clique problem. Specifically, we show that on an Erdös-Rényi graph G(n, 1 2 ), with high probability there is a feasible point for the degree-4 SOS relaxation of the clique problem with an objective value of Ω̃( √ n), so that the program cannot distinguish between a random graph and a random graph with a planted clique of size Õ( √ n). This bound is tight. We build on the works of Deshpande and Montanari and Meka et al., who give lower bounds of Ω̃(n) and Ω̃(n) respectively. We improve on their results by making a perturbation to the SDP solution proposed in their work, then showing that this perturbation remains PSD as the objective value approaches Ω̃(n). In an independent work, Hopkins, Kothari and Potechin [HKP15] have obtained a similar lower bound for the degree-4 SOS relaxation. UC Berkeley, [email protected]. Supported by NSF Career Award, NSF CCF-1407779 and the Alfred. P. Sloan Fellowship. UC Berkeley, [email protected]. Supported by an NSF Graduate Research Fellowship (NSF award no 1106400).