A bound for the number of different basic solutions generated by the simplex method | Mathematical Programming Skip to main content
Springer Nature Link
Log in

A bound for the number of different basic solutions generated by the simplex method

  • Short Communication
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems (LP) having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the positive elements of primal basic feasible solutions. When the problem is primal nondegenerate, it becomes a bound for the number of iterations. The result includes strong polynomiality for Markov Decision Problem by Ye (http://www.stanford.edu/~yyye/simplexmdp1.pdf, 2010) and utilize its analysis. We also apply our result to an LP whose constraint matrix is totally unimodular and a constant vector b of constraints is integral.

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
¥17,985 /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 includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dantzig G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)

    MATH  Google Scholar 

  2. Klee V., Minty G.J.: How good is the simplex method. In: Shisha, O. (eds) Inequalities III, Academic Press, New York (1972)

  3. Ye, Y.: The simplex and policy-iteration methods are strongly polynomial for the Markov Decision Problem with a fixed discount rate. Technical paper. http://www.stanford.edu/~yyye/simplexmdp1.pdf (2010) accessed 24 June 2011

Download references

Author information

Authors and Affiliations

  1. Graduate School of Decision Science and Technology, Tokyo Institute of Technology, 2-12-1-W9-62, Ookayama, Meguro-ku, Tokyo, 152-8552, Japan

    Tomonari Kitahara

  2. Graduate School of Decision Science and Technology, Tokyo Institute of Technology, 2-12-1-W9-58, Ookayama, Meguro-ku, Tokyo, 152-8552, Japan

    Shinji Mizuno

Authors
  1. Tomonari Kitahara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shinji Mizuno
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomonari Kitahara.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kitahara, T., Mizuno, S. A bound for the number of different basic solutions generated by the simplex method. Math. Program. 137, 579–586 (2013). https://doi.org/10.1007/s10107-011-0482-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-011-0482-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation