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 12 , 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 Θ̃( √ n), in sharp contrast to the classical information-theoretic threshold of Θ(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. ∗Supported by a Fannie & John Hertz Foundation Fellowship, a NSF Graduate Research Fellowship, and a Future of Life Institute grant. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 69 (2017)