Advertisement

A Mathematical Programming Approach to Marker-Assisted Gene Pyramiding

  • Stefan Canzar
  • Mohammed El-Kebir
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6833)

Abstract

In the crossing schedule optimization problem we are given an initial set of parental genotypes and a desired genotype, the ideotype. The task is to schedule crossings of individuals such that the number of generations, the number of crossings, and the required populations size are minimized. We present for the first time a mathematical model for the general problem variant and show that the problem is \(\mathcal{NP}\)-hard and even hard to approximate. On the positive side, we present a mixed integer programming formulation that exploits the intrinsic combinatorial structure of the problem. We are able to solve a real-world instance to provable optimality in less than 2 seconds, which was not possible with earlier methods.

Keywords

Source Node Directed Acyclic Graph Mixed Integer Linear Program Target Node Parental Genotype 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bradley, S.P., Hax, A.C., Magnanti, T.L.: Applied Mathematical Programming. Addison-Wesley, Reading (1977)Google Scholar
  2. 2.
    Brown, J., Caligari, P.: Introduction to Plant Breeding. Wiley-Blackwell (2008)Google Scholar
  3. 3.
    Collard, B.C.Y., Mackill, D.J.: Marker-assisted selection: an approach for precision plant breeding in the twenty-first century. Phil. Trans. R. Soc. B 363(1491), 557–572 (2008)CrossRefGoogle Scholar
  4. 4.
    Dekkers, J.C.M., Hospital, F.: The use of molecular genetics in the improvement of agricultural populations. Nature Reviews Genetics 3, 22–32 (2002)CrossRefGoogle Scholar
  5. 5.
    Haldane, J.B.S.: The combination of linkage values and the calculation of distances between the loci of linked factors. Journal of Genetics 8, 299–309 (1919)CrossRefGoogle Scholar
  6. 6.
    Ishii, T., Yonezawa, K.: Optimization of the marker-based procedures for pyramiding genes from multiple donor lines: I. Schedule of crossing between the donor lines. Crop Science 47, 537–546 (2007)CrossRefGoogle Scholar
  7. 7.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  8. 8.
    Moose, S.P., Mumm, R.H.: Molecular plant breeding as the foundation for 21st century crop improvement. Plant Physiology 147, 969–977 (2008)CrossRefGoogle Scholar
  9. 9.
    Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proc. 29th ACM Symp. on Theory of Computing, pp. 475–484 (1997)Google Scholar
  10. 10.
    Servin, B., Martin, O.C., Mézard, M., Hospital, F.: Toward a theory of marker-assisted gene pyramiding. Genetics 168(1), 513–523 (2004)CrossRefGoogle Scholar
  11. 11.
    Shifriss, C., Pilowsky, M., Zacks, J.M.: Resistance to Leveillula Taurica mildew (=Oidiopsis taurica) in Capsicum annuum. Phytoparasitica 20(4), 279–283 (1992)CrossRefGoogle Scholar
  12. 12.
    Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation and Application. Krieger Pub. Co. (1986)Google Scholar
  13. 13.
    Wolsey, L.A.: Integer programming. Wiley-Interscience series in discrete mathematics and optimization. Wiley, Chichester (1998)zbMATHGoogle Scholar
  14. 14.
    Ye, G., Smith, K.F.: Marker-assisted gene pyramiding for inbred line development: Basic principles and practical guidelines. International Journal of Plant Breeding 2(1), 1–10 (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Stefan Canzar
    • 1
  • Mohammed El-Kebir
    • 1
    • 2
  1. 1.Centrum Wiskunde & InformaticaLife Sciences GroupAmsterdamThe Netherlands
  2. 2.Centre for Integrative Bioinformatics VU (IBIVU)VU University AmsterdamAmsterdamThe Netherlands

Personalised recommendations