Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Polynomial algorithms for linear programming over the algebraic numbers

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We derive a bound on the computational complexity of linear programs whose coefficients are real algebraic numbers. Key to this result is a notion of problem size that is analogous in function to the binary size of a rational-number problem. We also view the coefficients of a linear program as members of a finite algebraic extension of the rational numbers. The degree of this extension is an upper bound on the degree of any algebraic number that can occur during the course of the algorithm, and in this sense can be viewed as a supplementary measure of problem dimension. Working under an arithmetic model of computation, and making use of a tool for obtaining upper and lower bounds on polynomial functions of algebraic numbers, we derive an algorithm based on the ellipsoid method that runs in time bounded by a polynomial in the dimension, degree, and size of the linear program. Similar results hold under a rational number model of computation, given a suitable binary encoding of the problem input.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Adler and P. A. Beling, Polynomial Algorithms for LP over a Subring of the Algebraic Integers with Applications to LP with Circulant Matrices,Mathematical Programming 57 (1992), 121–143.

    Google Scholar 

  2. I. Adler and P. A. Beling, Turing Algorithms for Linear Programming over the Algebraic Numbers, manuscript, September, 1992.

  3. P. A. Beling, Linear Programming over the Algebraic Numbers, Ph.D. dissertation, University of California, Berkeley, 1991.

    Google Scholar 

  4. L. Blum, M. Shub, and S. Smale, On a Theory of Computation and Complexity over the Real Numbers; NP-completeness, Recursive Functions and Universal Machines,Bulletin of the AMS 21(1) (1989), 1–46.

    Google Scholar 

  5. M. Grotschel, L. Lovasz, and A. Schrijver,Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988.

    Google Scholar 

  6. K. Ireland and M. Rosen,A Classical Introduction to modern Number Theory, Springer-Verlag, New York, 1972.

    Google Scholar 

  7. N. Karmarkar, A New Polynomial Time Algorithm for Linear Programming,Combinatorica 4 (1984), 373–395.

    Google Scholar 

  8. L. Khachiyan, A Polynomial Algorithm in Linear Programming,Soviet Mathematics Doklady 20 (1979), 191–194.

    Google Scholar 

  9. L. Lovasz,An Algorithmic Theory of Numbers, Graphs and Covexity, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1986.

    Google Scholar 

  10. N. Megiddo, Towards a Genuinely Polynomial Algorithm for Linear Programming,SIAM Journal on Computing 12(2) (1983), 347–353.

    Google Scholar 

  11. N. Megiddo, Linear Programming in Linear Time when the Dimension is Fixed,Journal of the Association for Computing Machinery 31 (1984), 114–127.

    Google Scholar 

  12. N. Megiddo, On Solving the Linear Programming Problem Approximately,Contemporary mathematics, Vol. 114, The American Mathematical Society, Providence, RI, 1990, pp. 35–50.

    Google Scholar 

  13. M. Mignotte, Some Useful Bounds, in: B. Buchberger, G. E. Collins, and R. Loos (eds.),Computer Algebra, Springer-Verlag, Wien, 1983, pp. 259–263.

    Google Scholar 

  14. C. Norton, S. Plotkin, and E. Tardos, using Separation Algorithms in Fixed Dimension,Proceedings of the 1st ACM/SIAM Symposium on Discrete Algorithms, 1990, pp. 377–387.

  15. C. H. Papadimitriou and K. Steiglitz,Combinatorial Optimization, Prentice-Hall, Englewood Cliffs, NJ, 1982.

    Google Scholar 

  16. H. Pollard and H. G. Diamond,The Theory of Algebraic numbers, 2nd edn., The Mathematical Association of America, Washington, DC, 1975.

    Google Scholar 

  17. A. B. Shidlovskii,Transcendental Numbers, de Gruyter, Berlin, 1989.

    Google Scholar 

  18. I. N. Stewart and D. O. Tall,Algebraic Number Theory, Chapman & Hall, New York, 1987.

    Google Scholar 

  19. E. Tardos, A Strongly Polynomial Algorithm To Solve Combinatorial Linear Programs,Opterations Research 34 (1986), 250–256.

    Google Scholar 

  20. J. F. Traub and H. Wozniakowski, Complexity of Linear Programming,Operations Research Letters 1 (1982), 59–62.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial Engineering and Operations Research, University of California, 94720, Berkeley, CA, USA

    I. Adler

  2. Department of Systems Engineering, University of Virginia, 22903, Charlottesville, VA, USA

    P. A. Beling

Authors
  1. I. Adler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. A. Beling
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Nimrod Megiddo.

This research was founded by the National Science Foundation under Grant DMS88-10192.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Adler, I., Beling, P.A. Polynomial algorithms for linear programming over the algebraic numbers. Algorithmica 12, 436–457 (1994). https://doi.org/10.1007/BF01188714

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01188714

Key words

Search

Navigation