Abstract
This paper studies the passage from a linear to an integer program using tools provided by test sets and cutting planes. The first half of the paper examines the process by which the secondary polytope Σ(A) associated with a matrix A refines to the state polytope St(A). In the second half of the paper, we show how certain elements in a test set can be used to provide inequalities that are valid for the optimal solutions of a 0/1 integer program. We close with complexity results for certain integer programs, apparent from their test sets.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
D. Bayer and I. Morrison, Gröbner bases and geometric invariant theory I, Journal of Symbolic Computation, 6, 1988, 209–217.
MACAULAY: a computer algebra system for algebraic geometry, available by anonymous ftp from zariski.harvard.edu.
L.J. Billera and B. Sturmfels, Fiber polytopes, Annals of Mathematics, 135, 1992, 527–549.
L.J. Billera, P. Filliman and B, Sturmfels, Constructions and complexity of secondary polytopes, Advances in Mathematics, 83, 1990, 155–179.
P. Conti and C. Traverso, Buchberger algorithm and integer programming, Proceedings AAECC-9 (New Orleans), Springer Verlag LNCS 539, 1991, 130–139.
I.M. Gel'fand, M. Kapranov and A. Zelevinsky, Multidimensional Determinants, Discriminants and Resultants, Birkhäuser, Boston, 1994.
M. Grötschel, L. Lovász, and A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics 2, Springer, Berlin, 1988; Second edition 1993.
M. Kapranov, B. Sturmfels and A.V. Zelevinsky, Chow polytopes and general resultants, Duke Math. Journal, Vol 67, 1992, 189–218.
T. Mora and L. Robbiano, The Gröbner fan of an ideal, Journal of Symbolic Computation, 6, 1988, 183–208.
A. Schrijver, Theory of Linear and Integer Programming, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York, 1986.
B. Sturmfels, Gröbner bases of toric varieties, Tôhoku Math. Journal, 43, 1991, 249–261.
B. Sturmfels, Gröbner Bases and Convex Polytopes, AMS University Lecture Series 8, Providence, RI, 1996.
B.Sturmfels and R.Thomas, Variation of cost functions in integer programming, Technical Report, 1994, Cornell University.
R. Thomas, A geometric Buchberger algorithm for integer programming, Mathematics of Operations Research, 20, 1995, 864–884.
A. Schulz, R. Weismantel and G. Ziegler, 0/1 integer programming: optimization and augmentation are equivalent, to appear in Proceedings of the European Symposium on Algorithms 95, “Lecture Notes in Computer Science”, Springer-Verlag, 1995.
R. Weismantel, “Hilbert bases and the facets of special knapsack polytopes”, Preprint SC 94-19, Konrad-Zuse-Zentrum Berlin, 1994.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Thomas, R.R., Weismantel, R. (1996). Test sets and inequalities for integer programs. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 1996. Lecture Notes in Computer Science, vol 1084. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61310-2_2
Download citation
DOI: https://doi.org/10.1007/3-540-61310-2_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61310-7
Online ISBN: 978-3-540-68453-4
eBook Packages: Springer Book Archive