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We consider the problem of identifying the densest k-node subgraph in a givengraph. We write this problem as an instance of rank-constrained cardinalityminimization and then relax using the nuclear and 11 norms. Although theoriginal combinatorial problem is NP-hard, we show that the densest k-subgraphcan be recovered from the solution of our convex relaxation for certain programinputs. In particular, we establish exact recovery in the case that the inputgraph contains a single planted clique plus noise in the form of corruptedadjacency relationships. We consider two constructions for this noise. In thefirst, noise is introduced by an adversary deterministically deleting edgeswithin the planted clique and placing diversionary edges. In the second, theseedge corruptions are performed at random. Analogous recovery guarantees foridentifying the densest subgraph of fixed size in a bipartite graph are alsoestablished, and results of numerical simulations for randomly generated graphsare included to demonstrate the efficacy of our algorithm.

Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation