  We consider the k-disjoint-clique problem. The input is an undirected graph G
in which the nodes represent data items, and edges indicate a similarity
between the corresponding items. The problem is to find within the graph k
disjoint cliques that cover the maximum number of nodes of G. This problem may
be understood as a general way to pose the classical `clustering' problem. In
clustering, one is given data items and a distance function, and one wishes to
partition the data into disjoint clusters of data items, such that the items in
each cluster are close to each other. Our formulation additionally allows
`noise' nodes to be present in the input data that are not part of any of the
cliques. The k-disjoint-clique problem is NP-hard, but we show that a convex
relaxation can solve it in polynomial time for input instances constructed in a
certain way. The input instances for which our algorithm finds the optimal
solution consist of k disjoint large cliques (called `planted cliques') that
are then obscured by noise edges and noise nodes inserted either at random or
by an adversary.
