Summary: Finding a large hidden clique in a random graph
Benny Sudakov §
We consider the following probabilistic model of a graph on n labeled vertices. First choose
a random graph G(n, 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 > cn0.5
, for any fixed c > 0, thus improving the
trivial case k > cn0.5
. The algorithm is based on the spectral properties of the graph.
A clique in a graph G is a set of vertices any two of which are connected by an edge. Let w(G)
denote the maximum number of vertices in a clique of G.
The problem of determining or estimating w(G) and that of finding a clique of maximum size in G
are fundamental problems in Theoretical Computer Science. The problem of computing w(G) is well