Finding a large hidden clique in a random graph

We consider the following probabilistic model of a graph on n labeled vertices. . First choose a random graph Gn ,1 r 2 ,and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kucera. In this paper we present an efficient algorithm for all k ) cn 0.5 , for ˇ 0.5 . 0.5 any fixed c ) 0, thus improving the trivial case k ) cn log n . The algorithm is based on the spectral properties of the graph. Q 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 457)466, 1998