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

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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.

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References

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

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  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

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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
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  2. Shinji Mizuno
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Correspondence to Tomonari Kitahara.

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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

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

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