Abstract
Benders decomposition is a well-known procedure for solving a combinatorial optimization problem by defining it in terms of a master problem and a slave problem. Its effectiveness relies, among other factors, on the possibility of synthesizing Benders cuts that rule out not only one, but a large class of trial values for the master problem. In turn, for the class of problems we consider (i.e., optimization plus constraint satisfaction) the possibility of separating the slave problem into several subproblems—i.e., problems exhibiting strong intra-relationships and weak inter-relationships—can be exploited for improving searching procedures efficiency. The notion of separation is typically given informally, or relying on syntactical aspects. This paper formally addresses the notion of slave problem separability by giving a semantic definition and exploring it from the computational point of view. Several examples of separable problems are provided, including some proving that a semantic notion of separability is much more helpful than a syntactic one. We show that separability can be formally characterized as equivalence of logical formulae, and prove the undecidability of the separability check problem. Finally, we show how there are cases where automated tools can still be used for checking subproblem separability.
Similar content being viewed by others
References
Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4, 238–252.
Börger, E., Gräedel, E., & Gurevich, Y. (1997). The classical decision problem. Perspectives in mathematical logic. Berlin: Springer.
Cadoli, M., Mancini, T., Micaletto, D., & Patrizi, F. (2006). Evaluating ASP and commercial solvers on the CSPLib. In Proceedings of the seventeenth European conference on artificial intelligence (ECAI 2006).
Cambazard, H., & Jussien, N. (2005). Integrating Benders decomposition within constraint programming. In Lecture notes in artificial intelligence : Vol. 3709. Proceedings of the eleventh international conference on principles and practice of constraint programming (CP 2005) (pp. 752–756). Berlin: Springer.
Dechter, R. (2003). Constraint processing. San Mateo: Morgan Kaufmann.
Fagin, R. (1974). Generalized first-order spectra and polynomial-time recognizable sets. In R. M. Karp (Ed.), Complexity of computation (pp. 43–74). Providence: Am. Math. Soc.
Hooker, J. (2000). Logic-based methods for optimization: combining optimization and constraint satisfaction (Chap. 19, pp. 389–422). New York: Wiley.
Hooker, J. N., & Ottosson, G. (2003). Logic-based Benders decomposition. Mathematical Programming, 96, 33–60.
Jain, V., & Grossmann, I. E. (2001). Algorithms for hybrid MILP/CP models for a class of optimization problems. INFORMS Journal on Computing, 13, 258–276.
Lau, K. F., & Dill, K. A. (1989). A lattice statistical mechanics model of the conformational and sequence spaces of proteins. Macromolecules, 22, 3986–3997.
Lenzerini, M. (2002). Data integration: A theoretical perspective. In Proceedings of the twentyfirst ACM SIGACT SIGMOD SIGART symposium on principles of database systems (PODS 2002) (pp. 233–246).
Papadimitriou, C. H. (1994). Computational complexity. Reading, MA: Addison Wesley.
Puget, J. F. (1998). A fast algorithm for the bound consistency of alldiff constraints. In Proceedings of the fifteenth national conference on artificial intelligence (AAAI’98) (pp. 359–366).
Quaife, A. (1992). Automated development of fundamental mathematical theories. Dordrecht: Kluwer Academic.
Smith, B. M., Stergiou, K., & Walsh, T. (2000). Using auxiliary variables and implied constraints to model non-binary problems. In AAAI/IAAI (pp. 182–187).
Van Hentenryck, P. (1999). The OPL optimization programming language. Cambridge: MIT Press.
Walsh, T. (2001). Permutation problems and channelling constraints. In Lecture notes in computer science : Vol. 2250. Proceedings of the eighth international conference on logic for programming, artificial intelligence and reasoning (LPAR 2001) (pp. 377–391). Berlin: Springer.
Wos, L. (1996). The automation of reasoning: An experimenter notebook with OTTER tutorial. San Diego: Academic Press.
Author information
Authors and Affiliations
Corresponding author
Additional information
An earlier version of this paper appeared in Proc. of 3rd International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (CP-AI-OR’06). Cork, Ireland. May, 2006.
Rights and permissions
About this article
Cite this article
Cadoli, M., Patrizi, F. On the separability of subproblems in Benders decompositions. Ann Oper Res 171, 27–43 (2009). https://doi.org/10.1007/s10479-008-0383-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-008-0383-5