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 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. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n). However, the analysis in the paper gives 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), and has a failure probability that is less than polynomially small.