Abstract
In interval propagation approaches to solving nonlinear constraints over reals it is common to build stronger propagators from systems of linear equations. This, as far as we are aware, is not pursued for integer finite domain propagation. In this paper we show how we use interval Gauss–Jordan elimination to build stronger propagators for an integer propagation solver. In a similar fashion we present an interval Fourier elimination preconditioning technique to generate redundant linear constraints from a system of linear inequalities. We show how to convert the resulting interval propagators into integer propagators. This allows us to use existing integer solvers. We give experiments that show how these preconditioning techniques can improve propagation performance on dense systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chiu, C. K., & Lee, J. H. M. (1995). Interval linear constraint solving using the preconditioned interval Gauss–Seidel method. In Twelfth international conference on logic programming (pp. 17–31). MIT Press.
Choi, C. W., Harvey, W., Lee, J. H. M., & Stuckey, P. J. (2006). Finite domain bounds consistency revisited. In Proceedings of the Australian conference on artificial intelligence 2006, LNCS (Vol. 4304, pp. 49–58). Springer.
ECLiPSe. http://eclipse.crosscoreop.com/eclipse/.
Feydy, T., & Stuckey, P. J. (2007). Propagating dense systems of integer linear equations. In Proceedings 22nd annual ACM symposium on applied computing (pp. 306–310). ACM Press.
Fourier, J. B. J. (1824). Reported in: Analyse de travaux de l’A cademie Royale des Sciences, pendant l’annee 1824, Partie Mathematique, Histoire de l’Academie Royale de Sciences de l’Institut de France 7 (1827) xlvii-lv. (Partial English translation in: D.A . Kohler, Translation of a Report by Fourier on his work on Linear Inequalities. Opsearch 10 (1973) 38-42.).
Goualard, F. (2008). Gaol reference manual. http://sourceforge.net/projects/gaol/.
Hansen, E., & Greenberg, R. I. (1983). An interval Newton method. Applied Mathematics and Computation, 12, 89–98.
Harvey, W., & Stuckey, P. J. (2003). Improving linear constraint propagation by changing constraint representation. Constraints, 8(2), 173–207.
Henderson, F., et al. (2007). The mercury language reference manual. http://www.mercury.cs.mu.oz.au.
Moore, R. (1966). Interval arithmetic. Prentice-Hall, Englewood Cliffs (NJ), USA.
Neumaier, A. (1990). Interval methods for systems of equations, Encyclopedia of Mathematics and its Applications (Vol. 37). Cambridge, UK: Cambridge University Press.
OPL Studio (2008). http://www.ilog.com/products/oplstudio/.
Shen, K., & Schimpf, J. (2005). Eplex: Harnessing mathematical programming solvers for constraint logic programming. In Proceedings of principles and practice of constraint programming, LNCS (Number 3709, pp. 622–636). Springer Verlag.
Solnon, C. (1997). Cooperation of LP solvers for solving MILPs. In Proceedings international conference on tools for artificial intelligence (pp. 240–247). IEEE Press.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Feydy, T., Stuckey, P.J. Propagating systems of dense linear integer constraints. Constraints 14, 235–253 (2009). https://doi.org/10.1007/s10601-008-9049-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10601-008-9049-9