Abstract
Recently, various algebraic integer programming (IP) solvers have been proposed based on the theory of Gröbner bases. The main difficulty of these solvers is the size of the Gröbner bases generated. In algorithms proposed so far, large Gröbner bases are generated by either introducing additional variables or by considering the generic IP problem IP A,C . Some improvements have been proposed such as Hosten and Sturmfels' method (GRIN) designed to avoid additional variables and Thomas' truncated Gröbner basis method which computes the reduced Gröbner basis for a specific IP problem IP A,C (b) (rather than its generalisation IP A,C ). In this paper we propose a new algebraic algorithm for solving IP problems. The new algorithm, called Minimised Geometric Buchberger Algorithm, combines Hosten and Sturmfels' GRIN and Thomas' truncated Gröbner basis method to compute the fundamental segments of an IP problem IP A,C directly in its original space and also the truncated Gröbner basis for a specific IP problem IP A,C (b). We have carried out experiments to compare this algorithm with others such as the geometric Buchberger algorithm, the truncated geometric Buchberger algorithm and the algorithm in GRIN. These experiments show that the new algorithm offers significant performance improvement.
Similar content being viewed by others
References
W.W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, Vol. 3 (American Mathematical Society, 1994).
B. Buchberger, Gröbner bases: an algorithm method in polynomial ideal theory, in: Multidimensional Systems Theory, ed. N.K. Bose (Reidel, Dordrecht, 1985) pp. 184–232.
P. Conti and C. Traverso, Buchberger algorithm and integer programming, in: Proceedings AAECC-9, New Orleans, Lecture Notes in Comput. Sci. 539 (Springer), pp. 130–139.
D. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms (Springer, New York, 1992).
R. Gebauer and H.M. Möller, On an installation of Buchberger's algorithm, Journal of Symbolic Computation 6 (1988) 275–286.
D.R. Grayson and M.E. Stillman, Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2.
A. Heck, Introduction to Maple, a Computer Algebra System (Springer, 1993).
S. Hosten and B. Sturmfels, Grin: An implementation of Gröbner bases for integer programming, in: Integer Programming and Combinatorial Optimization, eds. E. Balas, J. Clausen, Lecture Notes in Comput. Sci. 920 (Springer), pp. 207–276.
Q. Li, Y.K. Guo and T. Ida, A parallel algebraic approach towards integer programming, in: Proceedings of the IASTED International Conference on Parallel and Distributed Computing and Systems, Washington, DC (1997) pp. 59-64.
Q. Li, Y.K. Guo and T. Ida, Modelling integer programming with logic: Language and implementation, IEICE Trans. Fundamentals of Electronics, Communications and Computer Sciences, to appear.
G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley, New York, 1988).
A. Schrijver, Theory of Linear and Integer Programming (Wiley, 1986).
B. Sturmfels, Gröbner Bases and Convex Polytopes, Vol. 8 (American Mathematical Society, 1996).
S.R. Tayur, R.R. Thomas and N.R. Natrj, An algebraic geometry algorithm for scheduling in presence of setups and correlated demands, Mathematical Programming 69 (1995) 369–401.
R.R. Thomas, A geometric Buchberger algorithm for integer programming, Mathematics of Operations Research 20 (1995) 864–884.
R.R. Thomas and R. Weismantel, Truncated Gröbner bases for integer programming, Applicable Algebra in Engineering, Communication and Computing 8 (1997) 241–257.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Li, Q., Guo, Yk., Darlington, J. et al. Minimised Geometric Buchberger Algorithm for Integer Programming. Annals of Operations Research 108, 87–109 (2001). https://doi.org/10.1023/A:1016050826491
Issue Date:
DOI: https://doi.org/10.1023/A:1016050826491