Finding a large hidden clique in a random graph

: We consider the following probabilistic model of a graph on n labeled vertices. Z. First choose a random graph Gn,1#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 0.5 Z. 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. # 1998 John Wiley Sons, Inc. Random Struct. Alg., 13, 457#466, 1998 1. INTRODUCTION A clique in a graph G is a set of vertices, any two of which are connected by an Z. edge. Let wG denote the maximum number of vertices in a clique of G. Z. The problem of determining or estimating wG and that of finding a clique of m...