WABI 2006: Algorithms in Bioinformatics pp 310-320

# Approximation Algorithms for Bi-clustering Problems

• Lusheng Wang
• Yu Lin
• Xiaowen Liu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4175)

## Abstract

One of the main goals in the analysis of microarray data is to identify groups of genes and groups of experimental conditions (including environments, individuals and tissues), that exhibit similar expression patterns. This is the so-called bi-clustering problem. In this paper, we consider two variations of the bi-clustering problem: the Consensus Submatrix Problem and the Bottleneck Submatrix Problem. The input of the problems contains a m×n matrix A and integers l and k. The Consensus Submatrix Problem is to find a l×k submatrix with l<m and k<n and a consensus vector such that the sum of distance between all rows in the submatrix and the vector is minimized. The Bottleneck Submatrix Problem is to find a l×k submatrix with l<m and k<n, an integer d and a center vector such that the distance between every row in the submatrix and the vector is at most d and d is minimized. We show that both problems are NP-hard and give randomized approximation algorithms for special cases of the two problems. Using standard techniques, we can derandomize the algorithms to get polynomial time approximation schemes for the two problems. To our knowledge, this is the first time that approximation algorithms with guaranteed ratio are presented for microarray analysis.

## Keywords

Approximation Algorithm Gene Expression Data Fractional Solution Center Vector Consensus Score

## References

1. 1.
Stoughton, R.B.: Applications of DNA microarrays in biology. Annual Rev. Biochem. 74, 53–82 (2005)
2. 2.
Allison, D.B., Cui, X., Page, G.P., Sabripou, M.: Microarray data analysis: from disarray to consolidation and consensus. Nature Reviews Genetics 7, 55–65 (2006)
3. 3.
Tavazoie, S., Hughes, J.D., Campbell, M.J., Cho, R.J., Church, G.M.: Systematic determination of genetic network architecture. Nat. Genet. 22, 281–285 (1999)
4. 4.
Wu, F.X., Zhang, W.J., Kusalik, A.J.: A genetic K-means clustering algorithm applied to gene expression data. In: Xiang, Y., Chaib-draa, B. (eds.) Canadian AI 2003. LNCS(LNAI), vol. 2671, pp. 520–526. Springer, Heidelberg (2003)
5. 5.
Tamayo, P., Slonim, D., Mesirov, J., Zhu, Q., Kitareewan, S., Dmitrovsky, E., Lander, E.S., Golub, T.R.: Interpreting patterns of gene expression with self-organizing maps: methods and application to hematopoietic differentiation. Proc. Nat’l Acad. Sci. USA 96, 2907–2912 (1999)
6. 6.
Ressom, H., Wang, D., Natarajan, P.: Clustering gene expression data using adaptive double selforganizing map. Physiol. Genomics 14, 35–46 (2003)Google Scholar
7. 7.
Eisen, M.B., Spellman, P.T., Brown, P.O., Botstein, D.: Cluster analysis and display of genome-wide expression patterns. Proc. Nat’l Acad. Sci. USA 95, 14863–14868 (1998)
8. 8.
Iyer, V.R., Eisen, M.B., Ross, D.T., Schuler, G., Moore, T., Lee, J.C., Trent, J.M., Staudt, L.M., Hudson Jr., J., Boguski, M.S., Lashkari, D., Shalon, D., Botstein, D., Brown, P.O.: The transcriptional program in the response of human fibroblasts to serum. Science 283, 83–87 (1999)
9. 9.
Qin, J., Lewis, D.P., Noble, W.S.: Kernel hierarchical gene clustering from microarray expression data. Bioinformatics 19, 2097–2104 (2003)
10. 10.
Alter, O., Brown, P.O., Botstein, D.: Generalized singular value decomposition for comparative analysis of genome-scale expression data sets of two different organisms. Proc. Nat’l Acad. Sci. USA 100, 3351–3356 (2003)
11. 11.
Holter, N.S., Mitra, M., Maritan, A., Cieplak, M., Banavar, J.R., Fedoroff, N.V.: Fundamental patterns underlying gene expression profiles: simplicity from complexity. Proc. Nat’l Acad. Sci. USA 97, 8409–8414 (2000)
12. 12.
Li, K.C., Yan, M., Yuan, S.S.: A simple statistical model for depicting the cdc15-synchronized yeast cell-cycle regulated gene expression data. Statistica Sinica 12, 141–158 (2002)
13. 13.
Tjaden, B.: An approach for clustering gene expression data with error Information. BMC Bioinformatics 7, 17 (2006)
14. 14.
Mecham, B.H., Wetmore, D.Z., Szallasi, Z., Sadovsky, Y., Kohane, I., Mariani, T.J.: Increased measurement accuracy for sequence-verified microarray probes. Physiol. Genomics 18, 308–315 (2004)
15. 15.
Rocke, D.M., Dubin, B.: A Model for Measurement Error for Gene Expression Arrays. J. of Computational Biology 8(6), 557–569 (2001)
16. 16.
Draghici, S., Khatri, P., Eklund, A.C., Szallasi, Z.: Reliability and reproducibility issues in DNA microarray measurements. Trends in Genetics 22(2), 101–109 (2006)
17. 17.
Brody, J.P., Williams, B.A., Wold, B.J., Quake, S.R.: Significance and statistical errors in the analysis of DNA microarray data. Proc. Nat’l Acad. Sci. USA 99, 12975–12978 (2002)
18. 18.
Purdom, E., Holmes, S.P.: Error distribution for gene expression data. Statistical Applications in Genetics and Molecular Biology 4(1), 16 (2005)
19. 19.
Cho, H., Lee, J.K.: Bayesian hierarchical error model for analysis of gene expression data. Bioinformatics 20, 2016–2025 (2004)
20. 20.
Getz, G., Levine, E., Domany, E.: Coupled two–way clustering analysis of gene microarray data. Proc. Nat’l Acad. Sci. USA, 12079–12084 (2000)Google Scholar
21. 21.
Cheng, Y., Church, G.M.: Biclustering of expression data. In: Proc. 8th Conf. on Intelligent Systems for Molecular Biology ISMB 2000, pp. 93–103 (2000)Google Scholar
22. 22.
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)
23. 23.
Lonardi, S., Szpankowski, W., Yang, Q.: Finding biclusters by random projections. In: Sahinalp, S.C., Muthukrishnan, S.M., Dogrusoz, U. (eds.) CPM 2004. LNCS, vol. 3109, pp. 102–116. Springer, Heidelberg (2004)
24. 24.
Peeters, R.: The maximum edge biclique problem is NP-complete. Discrete Applied Mathematics 131(3), 651–654 (2003)
25. 25.
Li, M., Ma, B., Wang, L.: On the closest string and substring problems. J. ACM 49(2), 157–171 (2002)
26. 26.
Gillman, D.: A Chernoff bound for random walks on expander graphs. In: Proc. 34th Symp. on Foundations of Computer Science FOCS 1993. IEEE Computer Society Press, Los Alamitos (1993)Google Scholar
27. 27.
Arora, S., Karger, D., Karpinski, M.: Polynomial-time approximation schemes for dense instances of NP-hard problems. In: Proc. 27th ACM Symp. on Theory of Computing STOC 1995, pp. 284–293. ACM Press, New York (1995)

## Authors and Affiliations

• Lusheng Wang
• 1
• Yu Lin
• 1
• 2
• Xiaowen Liu
• 1
1. 1.Department of Computer ScienceCity University of Hong KongHong Kong
2. 2.Institute of Computing TechnologyChinese Academy of SciencesBeijingChina