Algorithms approaching the threshold for semi-random planted clique

We design new polynomial-time algorithms for recovering planted cliques in the semi-random graph model introduced by Feige and Kilian~\cite{FK01}. The previous best algorithms for this model succeed if the planted clique has size at least \(n^{2/3}\) in a graph with \(n\) vertices (Mehta, Mckenzie, Trevisan, 2019 and Charikar, Steinhardt, Valiant 2017). Our algorithms work for planted-clique sizes approaching \(n^{1/2}\) -- the information-theoretic threshold in the semi-random model~\cite{steinhardt2017does} and a conjectured computational threshold even in the easier fully-random model. This result comes close to resolving open questions by Feige and Steinhardt. Our algorithms are based on higher constant degree sum-of-squares relaxation and rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in \emph{unbalanced} bipartite Erd\H{o}s--R\'enyi random graphs into algorithms for semi-random planted clique. The use of a higher-constant degree sum-of-squares is essential in our setting: we prove a lower bound on the basic SDP for certifying bicliques that shows that the basic SDP cannot succeed for planted cliques of size $k =o(n^{2/3})$. We also provide some evidence that the information-computation trade-off of our current algorithms may be inherent by proving an average-case lower bound for unbalanced bicliques in the low-degree-polynomials model.