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Hardness and Approximation of the Asynchronous Border Minimization Problem

(Extended Abstract)

  • Conference paper
Theory and Applications of Models of Computation (TAMC 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7287))

Abstract

We study a combinatorial problem arising from the microarrays synthesis. The objective of the BMP is to place a set of sequences in the array and to find an embedding of these sequences into a common supersequence such that the sum of the “border length” is minimized. A variant of the problem, called P-BMP, is that the placement is given and the concern is simply to find the embedding.

Approximation algorithms have been proposed for the problem [21] but it is unknown whether the problem is NP-hard or not. In this paper, we give a comprehensive study of different variations of BMP by presenting NP-hardness proofs and improved approximation algorithms. We show that P-BMP, 1D-BMP, and BMP are all NP-hard. In contrast with the result in [21] that 1D-P-BMP is polynomial time solvable, the interesting implications include (i) the array dimension (1D or 2D) differentiates the complexity of P-BMP; (ii) for 1D array, whether placement is given differentiates the complexity of BMP; (iii) BMP is NP-hard regardless of the dimension of the array. Another contribution of the paper is improving the approximation for BMP from O(n 1/2 log2 n) to O(n 1/4 log2 n), where n is the total number of sequences.

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References

  1. Bartal, Y.: Probabilistic approximations of metric spaces and its algorithmic applications. In: FOCS, pp. 184–193 (1996)

    Google Scholar 

  2. Bonizzoni, P., Vedova, G.D.: The complexity of multiple sequence alignment with SP-score that is a metric. TCS 259(1-2), 63–79 (2001)

    Article  MATH  Google Scholar 

  3. de Carvalho Jr., S.A., Rahmann, S.: Improving the Layout of Oligonucleotide Microarrays: Pivot Partitioning. In: Bücher, P., Moret, B.M.E. (eds.) WABI 2006. LNCS (LNBI), vol. 4175, pp. 321–332. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  4. de Carvalho Jr., S.A., Rahmann, S.: Microarray layout as quadratic assignment problem. In: Proc. GCB, pp. 11–20 (2006)

    Google Scholar 

  5. de Carvalho Jr., S.A., Rahmann, S.: Improving the design of genechip arrays by combining placement and embedding. In: Proc. 6th CSB, pp. 54–63 (2007)

    Google Scholar 

  6. Chatterjee, M., Mohapatra, S., Ionan, A., Bawa, G., Ali-Fehmi, R., Wang, X., Nowak, J., Ye, B., Nahhas, F.A., Lu, K., Witkin, S.S., Fishman, D., Munkarah, A., Morris, R., Levin, N.K., Shirley, N.N., Tromp, G., Abrams, J., Draghici, S., Tainsky, M.A.: Diagnostic markers of ovarian cancer by high-throughput antigen cloning and detection on arrays. Cancer Research 66(2), 1181–1190 (2006)

    Article  Google Scholar 

  7. Cretich, M., Chiari, M.: Peptide Microarrays Methods and Protocols. Methods in Molecular Biology, vol. 570. Human Press (2009)

    Google Scholar 

  8. Ernvall, J., Katajainen, J., Penttonen, M.: NP-completeness of the hamming salesman problem. BIT Numerical Mathematics 25, 289–292 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: STOC, pp. 448–455 (2003)

    Google Scholar 

  10. Feng, D.F., Doolittle, R.F.: Approximation algorithms for multiple sequence alignment. TCS 182(1), 233–244 (1987)

    Google Scholar 

  11. Fodor, S., Read, J., Pirrung, M., Stryer, L., Lu, A., Solas, D.: Light-directed, spatially addressable parallel chemical synthesis. Science 251(4995), 767–773 (1991)

    Article  Google Scholar 

  12. Gerhold, D., Rushmore, T., Caskey, C.T.: DNA chips: promising toys have become powerful tools. Trends in Biochemical Sciences 24(5), 168–173 (1999)

    Article  Google Scholar 

  13. Gusfield, D.: Efficient methods for multiple sequence alignment with guaranteed error bounds. Bulletin of Mathematical Biology 55(1), 141–154 (1993)

    MathSciNet  MATH  Google Scholar 

  14. Hannenhalli, S., Hubell, E., Lipshutz, R., Pevzner, P.A.: Combinatorial algorithms for design of DNA arrays. Adv. in Biochem. Eng./Biotech. 77, 1–19 (2002)

    Article  Google Scholar 

  15. Kaderali, L., Schliep, A.: Selecting signature oligonucleotides to identify organisms using DNA arrays. Bioinformatics 18, 1340–1349 (2002)

    Article  Google Scholar 

  16. Kahng, A.B., Mandoiu, I.I., Pevzner, P.A., Reda, S., Zelikovsky, A.: Scalable heuristics for design of DNA probe arrays. JCB 11(2/3), 429–447 (2004); Preliminary versions in WABI 2002 and RECOMB 2003

    Google Scholar 

  17. Kahng, A.B., Mandoiu, I.I., Reda, S., Xu, X., Zelikovsky, A.: Computer-aided optimization of DNA array design and manufacturing. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25(2), 305–320 (2006)

    Article  Google Scholar 

  18. Kasif, S., Weng, Z., Detri, A., Beigel, R., De Lisi, C.: A computational framework for optimal masking in the synthesis of oligonucleotide microarrays. Nucleic Acids Research 30(20), e106 (2002)

    Article  Google Scholar 

  19. Kundeti, V., Rajasekaran, S.: On the hardness of the border length minimization problem. In: BIBE, pp. 248–253 (2009)

    Google Scholar 

  20. Kundeti, V., Rajasekaran, S., Dinh, H.: On the border length minimization problem (BLMP) on a square array. CoRR, abs/1003.2839 (2010)

    Google Scholar 

  21. Li, C.Y., Wong, P.W.H., Xin, Q., Yung, F.C.C.: Approximating Border Length for DNA Microarray Synthesis. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 410–422. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  22. Li, F., Stormo, G.: Selection of optimal DNA oligos for gene expression arrays. Bioinformatics 17(11), 1067–1076 (2001)

    Article  Google Scholar 

  23. Melle, C., Ernst, G., Schimmel, B., Bleul, A., Koscielny, S., Wiesner, A., Bogumil, R., Möller, U., Osterloh, D., Halbhuber, K.-J., von Eggeling, F.: A technical triade for proteomic identification and characterization of cancer biomarkers. Cancer Research 64(12), 4099–4104 (2004)

    Article  Google Scholar 

  24. Rahmann, S.: The shortest common supersequence problem in a microarray production setting. Bioinformatics 19(suppl.2), 156–161 (2003)

    Google Scholar 

  25. Räihä, K.-J.: The shortest common supersequence problem over binary alphabet is NP-complete. Theoretical Computer Science 16(2), 187–198 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  26. Reinert, K., Lenhof, H.P., Mutzel, P., Mehlhorn, K., Kececioglu, J.D.: A branch-and-cut algorithm for multiple sequence alignment. In: RECOMB, pp. 241–250 (1997)

    Google Scholar 

  27. Slonim, D.K., Tamayo, P., Mesirov, J.P., Golub, T.R., Lander, E.S.: Class prediction and discovery using gene expression data. In: RECOMB, pp. 263–272 (2000)

    Google Scholar 

  28. Sung, W.K., Lee, W.H.: Fast and accurate probe selection algorithm for large genomes. In: Proc. 2nd CSB, pp. 65–74 (2003)

    Google Scholar 

  29. Welsh, J., Sapinoso, L., Kern, S., Brown, D., Liu, T., Bauskin, A., Ward, R., Hawkins, N., Quinn, D., Russell, P., Sutherland, R., Breit, S., Moskaluk, C., Frierson Jr., H., Hampton, G.: Large-scale delineation of secreted protein biomarkers overexpressed in cancer tissue and serum. PNAS 100(6), 3410–3415 (2003)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Communications & Networking, Aalto University School of Electrical Engineering, Aalto, Finland

    Alexandru Popa

  2. Department of Computer Science, University of Liverpool, UK

    Prudence W. H. Wong & Fencol C. C. Yung

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  1. Alexandru Popa
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  2. Prudence W. H. Wong
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  3. Fencol C. C. Yung
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Editors and Affiliations

  1. Department of Computer Science and Engineering, Resource Planning and Generation, Indian Institute of Technology Kanpur, 208016, Kanpur, India

    Manindra Agrawal

  2. Department of Pure Mathematics, Leeds, University of Leeds, LS2 9JT, UK

    S. Barry Cooper

  3. Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, 100190, Beijing, P.R. China

    Angsheng Li

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Popa, A., Wong, P.W.H., Yung, F.C.C. (2012). Hardness and Approximation of the Asynchronous Border Minimization Problem. In: Agrawal, M., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2012. Lecture Notes in Computer Science, vol 7287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29952-0_20

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  • DOI: https://doi.org/10.1007/978-3-642-29952-0_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29951-3

  • Online ISBN: 978-3-642-29952-0

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