Finding hidden cliques in linear time with high probability

We are given a graph G with n vertices, where a random subset of κ 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 κ-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 κ = c√n for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron [14]. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n²). However, the analysis in [14] 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(n²), and has a failure probability that is less than polynomially small.