Abstract
We consider the problems of finding a maximum clique in a graph and finding a maximum-edge biclique in a bipartite graph. Both problems are NP-hard. We write both problems as matrix-rank minimization and then relax them using the nuclear norm. This technique, which may be regarded as a generalization of compressive sensing, has recently been shown to be an effective way to solve rank optimization problems. In the special case that the input graph has a planted clique or biclique (i.e., a single large clique or biclique plus diversionary edges), our algorithm successfully provides an exact solution to the original instance. For each problem, we provide two analyses of when our algorithm succeeds. In the first analysis, the diversionary edges are placed by an adversary. In the second, they are placed at random. In the case of random edges for the planted clique problem, we obtain the same bound as Alon, Krivelevich and Sudakov as well as Feige and Krauthgamer, but we use different techniques.
Similar content being viewed by others
References
Ackerman, M., Ben-David, S.: Which data sets are ‘clusterable’?—A theoretical study of clusterability (2008)
Alon N., Krivelevich M., Sudakov B.: Finding a large hidden clique in a random graph. Random Struct. Algorithms 13, 457–466 (1998)
Candès, E. J., Recht, B.: Exact matrix completion via convex optimization. Available from http://arxiv.org/abs/0805.4471, May 2008
Candès E.J., Romberg J., Tao T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)
Donoho D.L.: Compressed Sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)
Feige U., Krauthgamer R.: Finding and certifying a large hidden clique in a semirandom graph. Random Struct. Algorithms 16(2), 195–208 (2000)
Füredi Z., Komlós J.: The eigenvalues of random symmetric matrices. Combinatorica 1(3), 233–241 (1981)
Garey M.R., Johnson D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)
Geman S.: A limit theorem for the norm of random matrices. Ann. Probab. 8(2), 252–261 (1980)
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)
Gillis, N., Glineur, F.: Nonnegative factorization and the maximum edge biclique problem. Available from http://arxiv.org/abs/0810.4225, (2008)
Håstad, J.: Clique is hard to approximate within \({n^{1-\epsilon}}\). In: 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), pp. 627–636. IEEE Computer Society Press, Los Alamitos, CA (1996)
Hoeffding W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58, 13–30 (1962)
Iasemidis L.D., Pardalos P., Sackellares J.C., Shiau D.-S.: Quadratic binary programming and dynamical system approach to determine the predictability of epileptic seizures. J. Comb. Optim. 5(1), 9–26 (2001)
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)
Mitzenmacher M., Upfal E.: Probability and Computing. Cambridge University Press, Cambridge (2005)
Peeters R.: The maximum edge biclique problem is NP-complete. Discrete Appl. Math. 131, 651–654 (2003)
Recht, B., Fazel, M., Parrilo, P.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. Available from http://arxiv.org/abs/0706.4138 (2007)
Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ (1997) (Reprint of the 1970 original, Princeton Paperbacks)
Watson G.A.: Characterization of the subdifferential of some matrix norms. Linear Algebra Appl. 170, 33–45 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by a Discovery Grant and Postgraduate Scholarship (Doctoral) from NSERC (Natural Science and Engineering Research Council of Canada) and the US Air Force Office of Scientific Research.
Rights and permissions
About this article
Cite this article
Ames, B.P.W., Vavasis, S.A. Nuclear norm minimization for the planted clique and biclique problems. Math. Program. 129, 69–89 (2011). https://doi.org/10.1007/s10107-011-0459-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-011-0459-x