Abstract
Given a bipartite graph G = (V 1,V 2,E) where edges take on both positive and negative weights from set \(\mathcal{S}\), the maximum weighted edge biclique problem, or \(\mathcal{S}\)-MWEB for short, asks to find a bipartite subgraph whose sum of edge weights is maximized. This problem has various applications in bioinformatics, machine learning and databases and its (in)approximability remains open. In this paper, we show that for a wide range of choices of \(\mathcal{S}\), specifically when \(\left| \frac{\min\mathcal{S}} {\max \mathcal{S}} \right| \in \Omega(\eta^{\delta-1/2}) \cap O(\eta^{1/2-\delta})\) (where η = max {|V 1|, |V 2|}, and δ ∈ (0,1/2]), no polynomial time algorithm can approximate \(\mathcal{S}\)-MWEB within a factor of n ε for some ε> 0 unless RP = NP. This hardness result gives justification of the heuristic approaches adopted for various applied problems in the aforementioned areas, and indicates that good approximation algorithms are unlikely to exist. Specifically, we give two applications by showing that: 1) finding statistically significant biclusters in the SAMBA model, proposed in [18] for the analysis of microarray data, is n ε-inapproximable; and 2) no polynomial time algorithm exists for the Minimum Description Length with Holes problem [4] unless RP = NP.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Machine Learning 56, 89–113 (2004)
Ben-Dor, A., Chor, B., Karp, R., Yakhini, Z.: Discovering local structure in gene expression data: The Order-Preserving Submatrix Problem. In: Proceedings of RECOMB 2002, pp. 49–57 (2002)
Bu, S.: The summarization of hierarchical data with exceptions. Master Thesis, Department of Computer Science, University of British Columbia (2004), http://www.cs.ubc.ca/grads/resources/thesis/Nov04/Shaofeng_Bu.pdf
Bu, S., Lakshmanan, L.V.S., Ng, R.T.: MDL Summarization with Holes. In: Proceedings of VLDB 2005, pp. 433–444 (2005)
Cheng, Y., Church, G.: Biclustering of expression data. In: Proceedings of ISMB 2000, pp. 93–103. AAAI Press, Menlo Park (2000)
Dawande, M., Keskinocak, P., Swaminathan, J.M., Tayur, S.: On Bipartite and multipartite clique problems. Journal of Algorithms 41(2), 388–403 (2001)
Feige, U.: Relations between average case complexity and approximation complexity. In: Proceedings of STOC 2002, pp. 534–543 (2002)
Feige, U., Kogan, S.: Hardness of approximation of the Balanced Complete Bipartite Subgraph problem. Technical Report MCS 2004-2004, The Weizmann Institute of Science (2004)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco (1979)
Fontana, P., Guha, S., Tan, J.: Recursive MDL Summarization and Approximation Algorithms (preprint, 2007)
Håstad, J.: Clique is hard to approximate within n 1 − ε. Acta Mathematica 182, 105–142 (1999)
Khot, S.: Ruling out PTAS for Graph Min-Bisection, Densest Subgraph and Bipartite Clique. In: Proceedings of FOCS 2004, pp. 136–145 (2004)
Madeira, S.C., Oliveira, A.L.: Biclustering algorithms for biological data analysis: a survey. IEEE/ACM Transactions on Computational Biology and Bioinformatics 1, 24–45 (2004)
Mishra, N., Ron, D., Swaminathan, R.: On finding large conjunctive clusters. In: Proceedings of COLT 2003, pp. 448–462 (2003)
Peeters, R.: The maximum edge biclique problem is NP-complete. Discrete Applied Mathematics 131, 651–654 (2003)
Swaminathan, J.M., Tayur, S.: Managing Broader Product Lines Through Delayed Differentiation Using Vanilla Boxes. Management Science 44, 161–172 (1998)
Tan, J., Chua, K., Zhang, L., Zhu, S.: Complexity study on clustering problems in microarray data analysis. Algorithmica 48(2), 203–219 (2007)
Tanay, A., Sharan, R., Shamir, R.: Discovering statistically significant biclusters in gene expression data. Bioinformatics 18(1), 136–144 (2002)
Zhang, L., Zhu, S.: A New Clustering Method for Microarray Data Analysis. In: Proceedings of CSB 2002, pp. 268–275 (2002)
Zuckerman, D.: Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number. In: Proceedings of STOC 2006, pp. 681–690 (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tan, J. (2008). Inapproximability of Maximum Weighted Edge Biclique and Its Applications. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2008. Lecture Notes in Computer Science, vol 4978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79228-4_25
Download citation
DOI: https://doi.org/10.1007/978-3-540-79228-4_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79227-7
Online ISBN: 978-3-540-79228-4
eBook Packages: Computer ScienceComputer Science (R0)