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Handbook of Combinatorial Optimization pp 533–572Cite as

Gröbner Bases in Integer Programming

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Abstract

Recently, application of the theory of Gröbner bases to integer programming has given rise to new tools and results in this field. Here we present this algebraic theory as the natural integer analog of the simplex approach to linear programming Although couched in algebra, the theory of Gröbner bases and its consequences for integer programming are intimately intertwined with polyhedral geometry and lattice arithmetic which are staples of the traditional approach to this subject.

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References

  1. W.W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, American Mathematical Society, Graduate Studies in Math., Vol. I II, 1994.

    Google Scholar 

  2. D. Bayer and I. Morrison, Gröbner bases and geometric invariant theory I, Journal of Symbolic Computation Vol. 6 (1988) pp. 209–217.

    Article  MATH  MathSciNet  Google Scholar 

  3. L.J. Billera, P. Filliman and B. Sturmfels, Constructions and complexity of secondary polytopes, Advances in MathematicsVol. 83 (1990) pp. 155–179.

    Article  MathSciNet  Google Scholar 

  4. L.J. Billera, I.M. Gel’fand and B. Sturmfels, Duality and minors of secondary polyhedra, Journal of Combinatorial Theory B Vol. 57 (1993) pp. 258–268.

    Article  MATH  MathSciNet  Google Scholar 

  5. L.J. Billera and B. Sturmfels, Fiber polytopes, Annals of Mathematics Vol. 135 (1992) pp. 527–549.

    Article  MATH  MathSciNet  Google Scholar 

  6. C.E. Blair and R.G. Jeroslow, The value function of an integer program, Mathematical Programming Vol. 23 (1982) pp. 237–273.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Buchberger, On Finding a Vector Space Basis of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal (German), Ph.D. Thesis, Univ of Innsbruck, Austria, 1965.

    Google Scholar 

  8. P. Conti and C. Traverso, Gröbner bases and integer programming, Proceedings AAECC-9 (New Orleans), Springer Verlag, LNCS Vol. 539 (1991) pp. 130–139.

    MathSciNet  Google Scholar 

  9. W. Cook, A.M.H. Gerards, A. Schrijver and É. Tardos, Sensitivity theorems in integer linear programming, Mathematical Programming Vol. 34 (1986) pp. 251–264.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Cornuejols, R. Urbaniak, R. Weismantel and L. Wolsey, Decomposition of integer programs and of generating sets, Fifth Annual European Symposium on Algorithms (ESA’97), Graz, Austria, 1997. To appear in LNCS, Springer-Verlag.

    Google Scholar 

  11. D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, Second edition, Springer-Verlag, New York, 1996.

    MATH  Google Scholar 

  12. J. de Loera, B. Sturmfels and R.R.Thomas, Gröbner bases and triangulations of the second hypersimplex, Combinatorica, Vol. 15 (1995) pp. 409–424.

    Article  MathSciNet  Google Scholar 

  13. P. Diaconis and B. Sturmfels, Algebraic algorithms for sampling from conditional distributionsAnnals of Statistics,to appear.

    Google Scholar 

  14. F. Di Biase and R. Urbanke, An algorithm to calculate the kernel of certain polynomial ring homomorphisms, Experimental Mathematics, Vol. 4 (1995) pp. 227–234.

    MATH  MathSciNet  Google Scholar 

  15. W. Fulton, Introduction to Toric Varieties, Princeton University Press, Princeton, New Jersey, 1993.

    MATH  Google Scholar 

  16. I.M. Gel’fand, M. Kapranov and A. Zelevinsky, Multidimensional Determinants, Discriminants and Resultants, Birkhäuser, Boston, 1994.

    Book  Google Scholar 

  17. J.E. Graver, On the foundations of linear and integer programming I, Mathematical Programming Vol. 8 (1975) pp. 207–226.

    Article  MathSciNet  Google Scholar 

  18. D. Grayson and M. StillmanMacaulay 2: a computer algebra system, available from http://www.math.uiuc.edu/dan/.

  19. M.Hayer and W.Hochstättler, personal communication.

    Google Scholar 

  20. S. Hosten, Degrees of Gröbner Bases of Integer Programs, Ph. D. Thesis, Cornell University, 1997.

    Google Scholar 

  21. S. Hosten and B. Sturmfels, GRIN: An implementation of Gröbner bases for integer programming, in Integer Programming and Combinatorial Optimization (E. Balas and J. Clausen eds.), LNCS Vol. 920 (1995) pp. 267–276.

    Google Scholar 

  22. T. Mora and L. Robbiano, The Gröbner fan of an ideal, Journal of Symbolic Computation Vol. 6 (1988) pp. 183–208.

    Article  MATH  MathSciNet  Google Scholar 

  23. H.E. Scarf, Neighborhood systems for production sets with indivisibilities, Econometrica Vol. 54 (1986) pp. 507–532.

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Schrijver, Theory of Linear and Integer Programming, WileyInterscience Series in Discrete Mathematics and Optimization, New York, 1986.

    MATH  Google Scholar 

  25. B. Sturmfels, Gröbner bases of toric varieties, Tôhoku Math. Journal Vol. 43 (1991) pp. 249–261.

    Article  MATH  MathSciNet  Google Scholar 

  26. B. Sturmfels, Asymptotic analysis of toric ideals, Memoirs of the Faculty of Science, Kyushu University Ser.A, Vol. 46 (1992) pp. 217–228.

    Article  MATH  MathSciNet  Google Scholar 

  27. B. Sturmfels, Gröbner Bases and Convex Polytopes, American Mathematical Society, Providence, RI, 1995.

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  29. S.R. Tayur, R.R. Thomas and N.R. Natraj, An algebraic geometry algorithm for scheduling in the presence of setups and correlated demands, Mathematical Programming, Vol. 69 (1995) pp. 369–401.

    MATH  MathSciNet  Google Scholar 

  30. R.R. Thomas, A geometric Buchberger algorithm for integer programming, Mathematics of Operations Research Vol. 20 (1995) pp. 864–884.

    Article  MATH  MathSciNet  Google Scholar 

  31. R.R.Thomas and R.Weismantel, Truncated Gröbner bases for integer programming, Applicable Algebra in Engineering, Communication and Computing,to appear.

    Google Scholar 

  32. R.Urbaniak, R.Weismantel and G.Ziegler, A variant of Buchberger’s algorithm for integer programming, SIAM J. on Discrete Mathematics, Vol. 1 (1997) pp. 96–108.

    Google Scholar 

  33. G. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, Springer Verlag, New York, 1995.

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

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

    Rekha R. Thomas

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  1. Rekha R. Thomas
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Editor information

Editors and Affiliations

  1. University of Minnesota, Minneapolis, USA

    Ding-Zhu Du

  2. University of Florida, Gainesville, USA

    Panos M. Pardalos

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© 1998 Kluwer Academic Publishers

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Thomas, R.R. (1998). Gröbner Bases in Integer Programming. In: Du, DZ., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0303-9_8

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  • DOI: https://doi.org/10.1007/978-1-4613-0303-9_8

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7987-4

  • Online ISBN: 978-1-4613-0303-9

  • eBook Packages: Springer Book Archive

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