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Variation of cost functions in integer programming

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Abstract

We study the problem of minimizingc · x subject toA · x =b. x ≥ 0 andx integral, for a fixed matrixA. Two cost functionsc andc′ are considered equivalent if they give the same optimal solutions for eachb. We construct a polytopeSt(A) whose normal cones are the equivalence classes. Explicit inequality presentations of these cones are given by the reduced Gröbner bases associated withA. The union of the reduced Gröbner bases asc varies (called the universal Gröbner basis) consists precisely of the edge directions ofSt(A). We present geometric algorithms for computingSt(A), the Graver basis, and the universal Gröbner basis.

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Authors and Affiliations

  1. Department of Mathematics, University of California ai Berkeley, 94720, Berkeley, CA, USA

    Bernd Sturmfels

  2. Department of Mathematics, Texas A&M University, 77843, College Station, TX, USA

    Rekha R. Thomas

Authors
  1. Bernd Sturmfels
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  2. Rekha R. Thomas
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Correspondence to Rekha R. Thomas.

Additional information

Research partially supported by the National Science Foundation and the David and Lucile Packard Foundation.

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Sturmfels, B., Thomas, R.R. Variation of cost functions in integer programming. Mathematical Programming 77, 357–387 (1997). https://doi.org/10.1007/BF02614622

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