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. 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. Our algorithms work for planted-clique sizes approaching n(1/2) - the information-theoretic threshold in the semi-random model and a conjectured computational threshold even in the easier fullyrandom model. This result comes close to resolving open questions by Feige and Steinhardt. To generate a graph in the semi-random planted-clique model, we first 1) plant a clique of size n in an n-vertex Erdos-Renyi graph with edge probability 1/2 and then adversarially add or delete an arbitrary number edges not touching the planted clique and delete any subset of edges going out of the planted clique. For every n > 0, we give an (1/epsilon) -time algorithm that recovers a clique of size.. in this model whenever n=n(1/2+epsilon) In fact, our algorithm computes, with high probability, a list of about../.. cliques of size.. that contains the planted clique. Our algorithms also extend to arbitrary edge probabilities.. and improve on the previous best guarantee whenever p <= 1 -n(-0.001). Our algorithms rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in unbalanced bipartite Erdos-Renyi random graphs into algorithms for semi-random planted clique. Analogous to the (conjecturally) optimal algorithms for the fully-random model, the previous best guarantees for semi-random planted clique correspond to spectral relaxations of biclique numbers based on eigenvalues of adjacency matrices. We construct an SDP lower bound that shows that the..2/3 threshold in prior works is an inherent limitation of these spectral relaxations. We go beyond this limitation by using higher-order sum-of-squares relaxations for biclique numbers. 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 lowdegree polynomial model.