Convex optimization for the densest subgraph and densest submatrix problems

We consider the densest $k$-subgraph problem, which seeks to identify the $k$-node subgraph of a given input graph with maximum number of edges. This problem is well-known to be NP-hard, by reduction to the maximum clique problem. We propose a new convex relaxation for the densest $k$-subgraph problem, based on a nuclear norm relaxation of a low-rank plus sparse decomposition of the adjacency matrices of $k$-node subgraphs to partially address this intractability. We establish that the densest $k$-subgraph can be recovered with high probability from the optimal solution of this convex relaxation if the input graph is randomly sampled from a distribution of random graphs constructed to contain an especially dense $k$-node subgraph with high probability. Specifically, the relaxation is exact when the edges of the input graph are added independently at random, with edges within a particular $k$-node subgraph added with higher probability than other edges in the graph. We provide a sufficient condition on the size of this subgraph $k$ and the expected density under which the optimal solution of the proposed relaxation recovers this $k$-node subgraph with high probability. Further, we propose a first-order method for solving this relaxation based on the alternating direction method of multipliers, and empirically confirm our predicted recovery thresholds using simulations involving randomly generated graphs, as well as graphs drawn from social and collaborative networks.