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Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation

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We consider the problem of identifying the densest k-node subgraph in a given graph. We write this problem as an instance of rank-constrained cardinality minimization and then relax using the nuclear norm and one norm. Although the original combinatorial problem is NP-hard, we show that the densest k-subgraph can be recovered from the solution of our convex relaxation for certain program inputs. In particular, we establish exact recovery in the case that the input graph contains a single planted clique plus noise in the form of corrupted adjacency relationships. We also establish analogous recovery guarantees for identifying the densest subgraph of fixed size in a bipartite graph, and include results of numerical simulations for randomly generated graphs to demonstrate the efficacy of our algorithm.

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This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation. We are also grateful to Inderjit Dhillon, Stephen Vavasis and Teng Zhang for their helpful comments and suggestions, and to Shiqian Ma for his insight and help implementing the ADMM algorithm used in Sect. 5.

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Correspondence to Brendan P. W. Ames.

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Communicated by Paul I. Barton.

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Ames, B.P.W. Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation. J Optim Theory Appl 167, 653–675 (2015).

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