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Algorithmica

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

Parameterized Complexity of Asynchronous Border Minimization

  • Robert Ganian
  • Martin Kronegger
  • Andreas Pfandler
  • Alexandru PopaEmail author
Article
  • 48 Downloads

Abstract

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.

Keywords

Parameterized complexity Fixed-parameter tractability NP-hard problem Microarrays 

Notes

Acknowledgements

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].

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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

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