Convex optimization for the planted k-disjoint-clique problem
For researchers in clustering and combinatorial optimization, this work provides a polynomial-time algorithm for a previously intractable problem under a specific generative model, though the result is incremental as it relies on planted clique assumptions.
The paper addresses the NP-hard k-disjoint-clique problem, a clustering formulation that allows noise nodes, and shows that a convex relaxation can solve it in polynomial time for instances with k disjoint large planted cliques obscured by noise edges and nodes, whether random or adversarial.
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.