Abstract
Context-free grammar constraints enforce that a sequence of variables forms a word in a language defined by a context-free grammar. The constraint has received a lot of attention in the last few years as it represents an effective and highly expressive modeling entity. Its application has been studied in the field of Constraint Programming, Mixed Integer Programming, and SAT to solve complex decision problems such as shift scheduling. In this theoretical study we demonstrate how the constraint can be linearized efficiently. In particular, we propose a lifted polytope which has only integer extreme points. Based on this result, for shift scheduling problems we prove the equivalence of Dantzig’s original set covering model and a lately introduced grammar-based model.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Achterberg, T., Berthold, T., Koch, T., Wolter, K.: Constraint Integer Programming: a New Approach to Integrate CP and MIP. In: Perron, L., Trick, M.A. (eds.) CPAIOR 2008. LNCS, vol. 5015, pp. 6–20. Springer, Heidelberg (2008)
Dantzig, G.: A comment on edieis traffic delay at toll booths. Operations Research 2, 339–341 (1954)
Coté, M.C., Gendron, B., Quimper, C.-G., Rousseau, L.-M.: Formal Languages for Integer Programming Modeling of Shift Scheduling Problems. Cirrelt Technical Report #CIRRELT-2007-64 (2007)
Ernst, A.T., Jiang, H., Krishnamoorthy, M., Owens, B., Sier, D.: An annotated bibliography of personnel scheduling and rostering. Annals of Operations Research 127, 21–144 (2004)
Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)
Kadioglu, S., Sellmann, M.: Efficient Context-Free Grammar Constraints. In: 23rd National Conference on Artificial Intelligence (AAAI), pp. 310–316 (2008)
Katsirelos, G., Narodytska, N., Walsh, T.: The Weighted CfgConstraint. In: Perron, L., Trick, M.A. (eds.) CPAIOR 2008. LNCS, vol. 5015, pp. 323–327. Springer, Heidelberg (2008)
Quimper, C.-G., Rousseau, L.-M.: Language Based Operators for Solving Shift Scheduling Problems. In: Seventh Metaheuristics International Conference (2007)
Quimper, C.-G., Walsh, T.: Global grammar constraints. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 751–755. Springer, Heidelberg (2006)
Quimper, C.-G., Walsh, T.: Decomposing global grammar constraints. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 590–604. Springer, Heidelberg (2007)
Sellmann, M.: The Theory of Grammar Constraints. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 530–544. Springer, Heidelberg (2006)
Yunes, T., Aron, I., Hooker, J.: An Integrated Solver for Optimization Problems Working Paper WPS-MAS-08-01, School of Business Administration, University of Miami (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pesant, G., Quimper, CG., Rousseau, LM., Sellmann, M. (2009). The Polytope of Context-Free Grammar Constraints. In: van Hoeve, WJ., Hooker, J.N. (eds) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. CPAIOR 2009. Lecture Notes in Computer Science, vol 5547. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01929-6_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-01929-6_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01928-9
Online ISBN: 978-3-642-01929-6
eBook Packages: Computer ScienceComputer Science (R0)