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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
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.
A. Barvinok and K. Woods: Short rational generating functions for lattice point problems, Journal of the American Math. Soc. 16 (2003), 957–979.
D. Cox, J. Little and D. O’shea: Using Algebraic Geometry, Graduate Texts in Mathematics 185, Springer-Verlag, New York, 1998.
L. Cox and J. George: Controlled rounding for tables with subtotals, Annals of Operations Research 20 (1989), 141–157.
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.
M. Develin and S. Sullivant: Markov bases of binary graph models, Annals of Combinatorics 7 (2003), 441–466; math.CO/0308280.
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.
D. Eiesenbud, D. Grayson, M. Stillman and B. Sturmfels: Mathematical Computations with Macaulay2, Algorithms and Computation in Mathematics, Vol. 8, Springer-Verlag, Heidelberg, 2001.
S. Hoşten and S. Sullivant: Gröbner bases and polyhedral geometry of reducible and cyclic models, J. Combinatorial Theory, Ser. A 100 277–301.
S. Hoşten and R. Thomas: Standard pairs and group relaxations in integer programming, J. Pure Appl. Algebra 139 (1999), 133–157.
S. Hoşten and R. Thomas: The associated primes of initial ideals of lattice ideals, Math. Res. Lett. 6 (1999), 83–97.
B. Huber and R. Hhomas: Computing Gröbner fans of toric ideals, Experimental Mathematics 9 (2000), 321–331.
A. Jensen: CaTS — A computer program for computing the Gröbner fan of a toric ideal, http://www.soopadoopa.dk/anders/cats/cats.html.
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.
M. Saito, B. Sturmfels and N. Takayama: Gröbner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics 6, Springer-Verlag, Berlin, 2000.
A. Schrijver: Theory of Linear and Integer Programming, Wiley Interscience, Chichester, 1986.
B. Sturmfels: Gröbner Bases and Convex Polytopes, University Lecture Series 8, American Mathematical Society, 1995.
B. Sturmfels and R. Thomas: Variation of cost functions in integer programming, Mathematical Programming 77 (1997), 357–387.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the National Science Foundation (DMS-0200729).