, Volume 81, Issue 1, pp 201–223 | Cite as

Parameterized Complexity of Asynchronous Border Minimization

  • Robert Ganian
  • Martin Kronegger
  • Andreas Pfandler
  • Alexandru PopaEmail author


Microarrays are research tools used in gene discovery as well as disease and cancer diagnostics. Two prominent but challenging problems related to microarrays are the Border Minimization Problem (BMP) and the Border Minimization Problem with given placement (P-BMP). The common task of these two problems is to create so-called probe sequences (essentially a string) in a microarray. Here, the goal of the former problem is to determine an assignment of each probe sequence to a unique cell of the array and afterwards to construct the sequences at their respective cells while minimizing the border length of the probes. In contrast, for the latter problem the assignment of the probes to the cells is already given. In this paper we investigate the parameterized complexity of the natural exhaustive variants of BMP and P-BMP, termed \(\text {BMP}^e\) and \(\text {P-BMP}^e\) respectively, under several natural parameters. We show that \(\text {BMP}^e\) and \(\text {P-BMP}^e\) are in FPT under the following two combinations of parameters: (1) the size of the alphabet (c), the maximum length of a sequence (string) in the input (\(\ell \)) and the number of rows of the microarray (r); and, (2) the size of the alphabet and the size of the border length (o). Furthermore, \(\text {P-BMP}^e\) is in FPT when parameterized by c and \(\ell \). We complement our tractability results with a number of corresponding hardness results.


Parameterized complexity Fixed-parameter tractability NP-hard problem Microarrays 



Supported by the Austrian Science Fund (FWF) under Grants S11408-N23, Y698, P25518-N23 and P26696, and the German Research Foundation (DFG) under Grant ER 738/2-1. Furthermore, by the Institutional research programme PN 1819 “Advanced IT resources to support digital transformation processes in the economy and society—RESINFO-TD” (2018), Project PN 1819-01-01 “Modeling, simulation, optimization of complex systems and decision support in new areas of IT&C research”, funded by the Ministry of Research and Innovation. Robert Ganian is also affiliated with FI MU, Brno, Czech Republic. We thank the anonymous Algorithmica reviewers for their very helpful comments and suggestions as well as the reviewers of an earlier version of this paper [17].


  1. 1.
    Andreev, K., Räcke, H.: Balanced graph partitioning. Theory Comput. Syst. 39(6), 929–939 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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 Res. 66(2), 1181–1190 (2006)CrossRefGoogle Scholar
  3. 3.
    Cretich, M., Chiari, M.: Peptide Microarrays: Methods and Protocols. Methods in Molecular Biology. Humana Press, New York (2009)CrossRefGoogle Scholar
  4. 4.
    de Carvalho Jr., S.A., Rahmann, S.: Improving the layout of oligonucleotide microarrays: pivot partitioning. In: Proceedings of WABI. Lecture Notes in Computer Science, vol. 4175, pp. 321–332. Springer (2006)Google Scholar
  5. 5.
    de Carvalho Jr., S.A., Rahmann, S.: Microarray layout as quadratic assignment problem. In: Proceedings of GCB. Lecture Notes in Informatics, vol. 83, pp. 11–20. GI (2006)Google Scholar
  6. 6.
    de Carvalho Jr., S.A., Rahmann, S.: Improving the design of genechip arrays by combining placement and embedding. In: Proceedings of CSB, vol. 6, pp. 54–63. Imperial College Press (2007)Google Scholar
  7. 7.
    de Carvalho Jr., S.A., Rahmann, S.: Microarray design and layout. Accessed 21 Feb 2017
  8. 8.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, Berlin (1999)CrossRefGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Feldmann, A.E.: Balanced partitions of grids and related graphs. Ph.D. thesis, ETH Zürich (2012)Google Scholar
  11. 11.
    Feldmann, A.E.: Fast balanced partitioning is hard even on grids and trees. Theor. Comput. Sci. 485, 61–68 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fellows, M.R., Lokshtanov, D., Misra, N., Rosamond, F.A., Saurabh, S.: Graph layout problems parameterized by vertex cover. In: Proceedings of ISAAC. Lecture Notes in Computer Science, vol. 5369, pp. 294–305. Springer (2008)Google Scholar
  13. 13.
    Flum, J., Grohe, M.: Describing parameterized complexity classes. Inf. Comput. 187(2), 291–319 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)Google Scholar
  15. 15.
    Fodor, S., Read, J.L., Pirrung, M.C., Stryer, L., Lu, A.T., Solas, D.: Light-directed, spatially addressable parallel chemical synthesis. Science 251(4995), 767–773 (1991)CrossRefGoogle Scholar
  16. 16.
    Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ganian, R., Kronegger, M., Pfandler, A., Popa, A.: Parameterized complexity of asynchronous border minimization. In: Proceedings of TAMC. Lecture Notes in Computer Science, vol. 9076, pp. 428–440. Springer (2015)Google Scholar
  18. 18.
    Gerhold, D., Rushmore, T., Caskey, C.T.: DNA chips: promising toys have become powerful tools. Trends Biochem. Sci. 24(5), 168–173 (1999)CrossRefGoogle Scholar
  19. 19.
    Hannenhalli, S., Hubell, E., Lipshutz, R., Pevzner, P.A.: Combinatorial algorithms for design of DNA arrays. Adv. Biochem. Eng. Biotechnol. 77, 1–19 (2002)Google Scholar
  20. 20.
    Kahng, A.B., Mandoiu, I.I., Pevzner, P.A., Reda, S., Zelikovsky, A.: Scalable heuristics for design of DNA probe arrays. J. Comput. Biol. 11(2/3), 429–447 (2004). Preliminary versions in WABI (2002) and RECOMB (2003)Google Scholar
  21. 21.
    Kahng, A.B., Mandoiu, I.I., Reda, S., Xu, X., Zelikovsky, A.: Computer-aided optimization of DNA array design and manufacturing. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 25(2), 305–320 (2006)CrossRefGoogle Scholar
  22. 22.
    Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kundeti, V., Rajasekaran, S.: On the hardness of the border length minimization problem. In: Proceedings of BIBE, pp. 248–253. IEEE Computer Society (2009)Google Scholar
  24. 24.
    Kundeti, V., Rajasekaran, S., Dinh, H.: Border length minimization problem on a square array. J. Comput. Biol. 21(6), 446–455 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lenstra, H.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Li, C.Y., Wong, P.W.H., Xin, Q., Yung, F.C.C.: Approximating border length for DNA microarray synthesis. In: Proceedings of TAMC. Lecture Notes in Computer Science, vol. 4978, pp. 410–422. Springer (2008)Google Scholar
  27. 27.
    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 Res. 64(12), 4099–4104 (2004)CrossRefGoogle Scholar
  28. 28.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2006)Google Scholar
  29. 29.
    Popa, A., Wong, P.W.H., Yung, F.C.C.: Hardness and approximation of the asynchronous border minimization problem (extended abstract). In: Proceedings of TAMC. Lecture Notes in Computer Science, vol. 7287, pp. 164–176. Springer (2012)Google Scholar
  30. 30.
    Slonim, D.K., Tamayo, P., Mesirov, J.P., Golub, T.R., Lander, E.S.: Class prediction and discovery using gene expression data. In: Proceedings of RECOMB, pp. 263–272. ACM (2000)Google Scholar
  31. 31.
    Welsh, J.B., Sapinoso, L.M., Kern, S.G., Brown, D.A., Liu, T., Bauskin, A.R., Ward, R.L., Hawkins, N.J., Quinn, D.I., Russell, P.J., Sutherland, R.L., Breit, S.N., Moskaluk, C.A., Frierson Jr., H.F., Hampton, G.M.: Large-scale delineation of secreted protein biomarkers overexpressed in cancer tissue and serum. Proc. Natl. Acad. Sci. 100(6), 3410–3415 (2003)CrossRefGoogle Scholar
  32. 32.
    Zhang, Y., Gladyshev, V.N.: High content of proteins containing 21st and 22nd amino acids, selenocysteine and pyrrolysine, in a symbiotic deltaproteobacterium of gutless worm olavius algarvensis. Nucleic Acids Res. 35(15), 4952–4963 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Robert Ganian
    • 2
  • Martin Kronegger
    • 1
  • Andreas Pfandler
    • 2
    • 4
  • Alexandru Popa
    • 3
    • 5
    Email author
  1. 1.Johannes Kepler University LinzLinzAustria
  2. 2.TU WienViennaAustria
  3. 3.University of BucharestBucharestRomania
  4. 4.University of SiegenSiegenGermany
  5. 5.National Institute for Research & Development in InformaticsBucharestRomania

Personalised recommendations