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The Graver Complexity of Integer Programming

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Abstract

In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver complexity of the incidence matrix of the complete bipartite graph K 3,m satisfies g(m) = Ω(2m), with g(m) ≥ 17·2m−3 –7 for every m > 3.

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References

  1. Aoki S., Takemura A.: Minimal basis for connected Markov chain over 3 × 3 × k contingency tables with fixed two-dimensional marginals. Aust. N. Z. J. Stat. 45, 229–249 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. De Loera J., Hemmecke R., Onn S., Weismantel R.: N-fold integer programming. Discrete Optim. 5, 231–241 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. De Loera J., Onn S.: All linear and integer programs are slim 3-way transportation programs. SIAM J. Optim. 17, 806–821 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. De Loera J., Onn S.: Markov bases of three-way tables are arbitrarily complicated. J. Symbolic Comput. 41, 173–181 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gordan P.: Über die Auflösung linearer Gleichungen mit reellen Coefficienten. Math. Ann. 6, 23–28 (1873)

    Article  MathSciNet  Google Scholar 

  6. Graver J.E.: On the foundations of linear and integer programming. Math. Program. 9, 207–226 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hemmecke R., Nairn K.A.: On the Gröbner complexity of matrices. J. Pure Appl. Algebra 213, 1558–1563 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Santos F., Sturmfels B.: Higher Lawrence configurations. J. Combin. Therory Ser. A 103, 151–164 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Sturmfels B.: Gröbner Bases and Convex Polytopes. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology, Technion, Haifa, 32000, Israel

    Yael Berstein & Shmuel Onn

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  1. Yael Berstein
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  2. Shmuel Onn
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Correspondence to Yael Berstein.

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Berstein, Y., Onn, S. The Graver Complexity of Integer Programming. Ann. Comb. 13, 289–296 (2009). https://doi.org/10.1007/s00026-009-0029-6

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  • DOI: https://doi.org/10.1007/s00026-009-0029-6

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