Finding hidden cliques in linear time with high probability

We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability 1/2. This random graph model is denoted G(n,1/2,k). The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov [3] uses spectral techniques to find the hidden clique with high probability when k = c√n for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron [12]. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n ²). However, the analysis in [12] gives a success probability of only 2/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time O(n ²) (that is, linear in the size of the input) and has a failure probability that tends to 0 as n tends to ∞. We develop this algorithm in the more general setting where the clique is replaced by a dense random graph.