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Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation

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  • Brendan P. W. Ames

    (University of Alabama)

Abstract

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.

Suggested Citation

  • Brendan P. W. Ames, 2015. "Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 653-675, November.
    • Handle: RePEc:spr:joptap:v:167:y:2015:i:2:d:10.1007_s10957-015-0777-x
    DOI: 10.1007/s10957-015-0777-x
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    References listed on IDEAS

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    1. Peeters, M.J.P., 2003. "The maximum edge biclique problem is NP-complete," Other publications TiSEM 3e340431-37b3-4bc5-9b14-9, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Polina Bombina & Brendan Ames, 2020. "Convex Optimization for the Densest Subgraph and Densest Submatrix Problems," SN Operations Research Forum, Springer, vol. 1(3), pages 1-24, September.

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