Convex optimization for the planted k-disjoint-clique problem

Abstract

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 inserted either at random or by an adversary, as well as additional nodes not belonging to any of the \(k\) planted cliques.