Finding Hidden Cliques of Size √N/e in Nearly Linear Time

Consider an Erdos-Renyi random graph in which each edge is present independently with probability , except for a subset of the vertices that form a clique (a completely connected subgraph). We consider the problem of identifying the clique, given a realization of such a random graph. The algorithm of Dekel et al. (ANALCO. SIAM, pp 67-75, 2011) provably identifies the clique in linear time, provided . Spectral methods can be shown to fail on cliques smaller than . In this paper we describe a nearly linear-time algorithm that succeeds with high probability for vertical bar C-N vertical bar (1 - epsilon) N/root e Delta for any epsilon > 0. This is the first algorithm that provably improves over spectral methods. We further generalize the hidden clique problem to other background graphs (the standard case corresponding to the complete graph on vertices). For large-girth regular graphs of degree (Delta + 1) we prove that so-called local algorithms succeed if vertical bar C-N vertical bar (1 - epsilon) N/root e Delta and fail if vertical bar C-N vertical bar (1 - epsilon) N/root e Delta.