Computing Kitahara–Mizuno’s bound on the number of basic feasible solutions generated with the simplex algorithm

Optimization Letters volume 12, pages933–943(2018)Cite this article

Abstract

This paper addresses an upper bound derived by Kitahara and Mizuno (Math Program A 137:579–586, 2013) on the number of basic feasible solutions of a linear program generated with the simplex algorithm. Their bound includes two parameters \(\delta \) and \(\gamma \), which are respectively the minimum and the maximum values of positive components in all basic feasible solutions. We show that \(\delta \) is NP-hard to determine while \(\gamma \) can be computed in polynomial time. We also report some numerical results using alternative parameters for \(\delta \) and \(\gamma \).

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Author notes

  1. Takahiro Tsuruda

    Present address: Cybozu, Inc., Tokyo, Japan

Affiliations

  1. Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba, Ibaraki, 305-8573, Japan

    Takahito Kuno, Yoshio Sano & Takahiro Tsuruda

Authors
  1. Takahito Kuno
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  2. Yoshio Sano
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  3. Takahiro Tsuruda
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Corresponding author

Correspondence to Takahito Kuno.

Additional information

Takahito Kuno: Partially supported by a Grant-in-Aid for Scientific Research (C) 16K00028 from the Japan Society for the Promotion of Sciences. Yoshio Sano: Partially supported by a Grant-in-Aid for Young Scientists (B) 15K20885 from the Japan Society for the Promotion of Sciences.

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Kuno, T., Sano, Y. & Tsuruda, T. Computing Kitahara–Mizuno’s bound on the number of basic feasible solutions generated with the simplex algorithm. Optim Lett 12, 933–943 (2018). https://doi.org/10.1007/s11590-018-1276-4

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Keywords

  • Linear programming
  • Simplex algorithm
  • Number of iterations
  • Basic feasible solutions
  • NP-hardness