Computing the integer programming gap

Abstract

We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed.

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References

  1. [1]

    L. Buzzigoli and A. Gusti: An algorithm to calculate the lower and upper bounds of the elements of an array given its marginals, in Statistical Data Protection Proceedings, Eurostat, Luxembourg (1999), pp. 131–147.

    Google Scholar 

  2. [2]

    A. Barvinok and J. E. Pommerscheim: A algorithmic theory of lattice points in polyhedra, in New Perspectives in Algebraic Combinatorics, MSRI Publications 38 (1999), pp. 91–147.

    Google Scholar 

  3. [3]

    A. Barvinok and K. Woods: Short rational generating functions for lattice point problems, Journal of the American Math. Soc. 16 (2003), 957–979.

    Article  MathSciNet  MATH  Google Scholar 

  4. [4]

    D. Cox, J. Little and D. O’shea: Using Algebraic Geometry, Graduate Texts in Mathematics 185, Springer-Verlag, New York, 1998.

    Google Scholar 

  5. [5]

    L. Cox and J. George: Controlled rounding for tables with subtotals, Annals of Operations Research 20 (1989), 141–157.

    Article  MathSciNet  MATH  Google Scholar 

  6. [6]

    J. A. De Loera, R. Hemmecke, J. Tauzer and R. Yoshida: Effective lattice point counting in rational convex polytopes, Symbolic Computation 38(4) (2004), 1273–1302.

    Article  MathSciNet  Google Scholar 

  7. [7]

    M. Develin and S. Sullivant: Markov bases of binary graph models, Annals of Combinatorics 7 (2003), 441–466; math.CO/0308280.

    Article  MathSciNet  MATH  Google Scholar 

  8. [8]

    A. Dobra and S. Fienberg: Bounds for cell entries in contingency tables given marginal totals and decomposable graphs, Proc. Natl. Acad. Sci. USA 97 (2000), 11885–11892.

    Article  MathSciNet  MATH  Google Scholar 

  9. [9]

    D. Eiesenbud, D. Grayson, M. Stillman and B. Sturmfels: Mathematical Computations with Macaulay2, Algorithms and Computation in Mathematics, Vol. 8, Springer-Verlag, Heidelberg, 2001.

    Google Scholar 

  10. [10]

    S. Hoşten and S. Sullivant: Gröbner bases and polyhedral geometry of reducible and cyclic models, J. Combinatorial Theory, Ser. A 100 277–301.

  11. [11]

    S. Hoşten and R. Thomas: Standard pairs and group relaxations in integer programming, J. Pure Appl. Algebra 139 (1999), 133–157.

    Article  MathSciNet  MATH  Google Scholar 

  12. [12]

    S. Hoşten and R. Thomas: The associated primes of initial ideals of lattice ideals, Math. Res. Lett. 6 (1999), 83–97.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    B. Huber and R. Hhomas: Computing Gröbner fans of toric ideals, Experimental Mathematics 9 (2000), 321–331.

    MathSciNet  MATH  Google Scholar 

  14. [14]

    A. Jensen: CaTS — A computer program for computing the Gröbner fan of a toric ideal, http://www.soopadoopa.dk/anders/cats/cats.html.

  15. [15]

    J. B. Lasserre: Duality and a Farkas lemma for integer programs, in Optimization: Structure and Applications (E. Hunt and C. E. M. Pearce, editors), Applied Optimization Series, Kluwer Academic Publishers, 2003.

  16. [16]

    M. Saito, B. Sturmfels and N. Takayama: Gröbner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics 6, Springer-Verlag, Berlin, 2000.

    Google Scholar 

  17. [17]

    A. Schrijver: Theory of Linear and Integer Programming, Wiley Interscience, Chichester, 1986.

    Google Scholar 

  18. [18]

    B. Sturmfels: Gröbner Bases and Convex Polytopes, University Lecture Series 8, American Mathematical Society, 1995.

  19. [19]

    B. Sturmfels and R. Thomas: Variation of cost functions in integer programming, Mathematical Programming 77 (1997), 357–387.

    MathSciNet  Google Scholar 

  20. [20]

    B. Sturmfels, R. Weismantel and G. Ziegler: Gröbner bases of lattices, corner polyhedra, and integer programming; Beiträge zur Algebra und Geometrie 36 (1995), 281–298.

    MathSciNet  MATH  Google Scholar 

  21. [21]

    R. R. Thomas: The structure of group relaxations, to appear in Handbook of Discrete Optimization (eds.: K. Aardal, G. Nemhauser, R. Weismantel), 2003, http://www.math.washington.edu/~thomas/articles.html.

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Correspondence to Serkan Hoşten.

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Partially supported by the National Science Foundation (DMS-0200729).

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Hoşten, S., Sturmfels, B. Computing the integer programming gap. Combinatorica 27, 367–382 (2007). https://doi.org/10.1007/s00493-007-2057-3

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Mathematics Subject Classification (2000)

  • 13P10
  • 90C10