Does robustness imply tractability? A lower bound for planted clique in the semi-random model

We consider a robust analog of the planted clique problem. In this analog, a set $S$ of vertices is chosen and all edges in $S$ are included; then, edges between $S$ and the rest of the graph are included with probability $\frac{1}{2}$, while edges not touching $S$ are allowed to vary arbitrarily. For this semi-random model, we show that the information-theoretic threshold for recovery is $\tilde{\mathrm{\Theta }}(\sqrt{n})$, in sharp contrast to the classical information-theoretic threshold of $\mathrm{\Theta }(\log (n))$. This matches the conjectured computational threshold for the classical planted clique problem, and thus raises the intriguing possibility that, once we require robustness, there is no computational-statistical gap for planted clique. Our lower bound involves establishing a result regarding the KL divergence of a family of perturbed Bernoulli distributions, which may be of independent interest.