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.
Similar content being viewed by others
References
Feige, U., Peleg, D., Kortsarz, G.: The dense k-subgraph problem. Algorithmica 29(3), 410–421 (2001)
Feige, U.: Relations between average case complexity and approximation complexity. In: Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pp. 534–543. ACM (2002)
Khot, S.: Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique. SIAM J. Comput. 36(4), 1025–1071 (2006)
Alon, N., Arora, S., Manokaran, R., Moshkovitz, D., Weinstein, O.: Inapproximability of densest \(\kappa \)-subgraph from average-case hardness (2011)
Ames, B., Vavasis, S.: Nuclear norm minimization for the planted clique and biclique problems. Math. Program. 129(1), 1–21 (2011)
Chandrasekaran, V., Sanghavi, S., Parrilo, P.A., Willsky, A.S.: Rank-sparsity incoherence for matrix decomposition. SIAM J. Optim. 21(2), 572–596 (2011)
Candès, E.J., Li, X., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM (JACM) 58(3), 11 (2011)
Chen, Y., Jalali, A., Sanghavi, S., Caramanis, C.: Low-rank matrix recovery from errors and erasures. IEEE Trans. Inf. Theory 59(7), 4324–4337 (2013)
Oymak, S., Hassibi, B.: Finding dense clusters via “low rank + sparse” decomposition. Arxiv preprint arXiv:1104.5186 (2011)
Chen, Y., Jalali, A., Sanghavi, S., Xu, H.: Clustering partially observed graphs via convex optimization. J. Mach. Learn. Res. 15(1), 2213–2238 (2014)
Chen, Y., Sanghavi, S., Xu, H.: Clustering sparse graphs. In: Advances in neural information processing systems, pp. 2204–2212 (2012)
Lawler, E.L.: Combinatorial optimization: networks and matroids. Courier Corporation (1976)
Karp, R.: Reducibility among combinatorial problems. Complex. Comput. Comput. 40(4), 85–103 (1972)
Doan, X.V., Vavasis, S.: Finding approximately rank-one submatrices with the nuclear norm and \(\ell _1\)-norm. SIAM J. Optim. 23(4), 2502–2540 (2013)
Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471–501 (2010)
Gilbert, A.C., Guha, S., Indyk, P., Muthukrishnan, S., Strauss, M.: Near-optimal sparse fourier representations via sampling. In: STOC ’02: Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pp. 152–161. ACM, New York, NY, USA (2002). doi:10.1145/509907.509933
Donoho, D.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)
Candès, E., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2) (2006)
Ames, B.: Robust convex relaxation for the planted clique and densest \(k\)-subgraph problems: additional proofs (2013). Available from http://bpames.people.ua.edu/uploads/3/9/0/0/39000767/dks_appendices.pdf
Kučera, L.: Expected complexity of graph partitioning problems. Discret. Appl. Math. 57(2), 193–212 (1995)
Alon, N., Krivelevich, M., Sudakov, B.: Finding a large hidden clique in a random graph. In: Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms, pp. 594–598. Society for Industrial and Applied Mathematics (1998)
Feige, U., Krauthgamer, R.: Finding and certifying a large hidden clique in a semirandom graph. Random Struct. Algorithms 16(2), 195–208 (2000)
McSherry, F.: Spectral partitioning of random graphs. In: Proceedings of the 42nd IEEE symposium on Foundations of Computer Science, pp. 529–537. IEEE Computer Society (2001)
Feige, U., Ron, D.: Finding hidden cliques in linear time. DMTCS Proc. 01, 189–204 (2010)
Dekel, Y., Gurel-Gurevich, O., Peres, Y.: Finding hidden cliques in linear time with high probability. Comb. Probab. Comput. 23(01), 29–49 (2014)
Deshpande, Y., Montanari, A.: Finding hidden cliques of size \(\sqrt{N/e}\) in nearly linear time. Found. Comput. Math. pp. 1–60 (2013)
Juels, A., Peinado, M.: Hiding cliques for cryptographic security. Des. Codes Crypt. 20(3), 269–280 (2000)
Alon, N., Andoni, A., Kaufman, T., Matulef, K., Rubinfeld, R., Xie, N.: Testing k-wise and almost k-wise independence. In: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pp. 496–505. ACM (2007)
Hazan, E., Krauthgamer, R.: How hard is it to approximate the best nash equilibrium? SIAM J. Comput. 40(1), 79–91 (2011)
Berthet, Q., Rigollet, P.: Computational lower bounds for sparse PCA. arXiv preprint arXiv:1304.0828 (2013)
Jerrum, M.: Large cliques elude the metropolis process. Random Struct. Algorithms 3(4), 347–359 (1992)
Feige, U., Krauthgamer, R.: The probable value of the lovász-schrijver relaxations for maximum independent set. SIAM J. Comput. 32(2), 345–370 (2003)
Nadakuditi, R.: On hard limits of eigen-analysis based planted clique detection. In: Statistical Signal Processing Workshop (SSP), 2012 IEEE, pp. 129–132. IEEE (2012)
Peeters, R.: The maximum edge biclique problem is NP-complete. Discret. Appl. Math. 131(3), 651–654 (2003)
Goerdt, A., Lanka, A.: An approximation hardness result for bipartite clique. In: Electronic Colloquium on Computational Complexity, Report, 48 (2004)
Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004)
Golub, G., Van Loan, C.: Matrix computations. Johns Hopkins University Press, Baltimore (1996)
Lugosi, G.: Concentration-measure inequalities (2009). Available from http://www.econ.upf.edu/~lugosi/anu.pdf
Tropp, J.: User-friendly tail bounds for sums of random matrices. Foundations of Computational Mathematics pp. 1–46 (2011)
Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)
Goldfarb, D., Ma, S., Scheinberg, K.: Fast alternating linearization methods for minimizing the sum of two convex functions. Math. Program. pp. 1–34 (2010)
Hong, M., Luo, Z.: On the linear convergence of the alternating direction method of multipliers. arXiv preprint arXiv:1208.3922 (2012)
Rohe, K., Qin, T., Fan, H.: The highest dimensional stochastic blockmodel with a regularized estimator. arXiv preprint arXiv:1206.2380 (2012)
Ames, B., Vavasis, S.: Convex optimization for the planted k-disjoint-clique problem. Math. Program. 143(1–2), 299–337 (2014)
Ames, B.: Guaranteed clustering and biclustering via semidefinite programming. Math. Program. 147(1–2), 429–465 (2014)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Paul I. Barton.
Rights and permissions
About this article
Cite this article
Ames, B.P.W. Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation. J Optim Theory Appl 167, 653–675 (2015). https://doi.org/10.1007/s10957-015-0777-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-015-0777-x