Abstract
The weighted constraint satisfaction problem (WCSP) occurs in the crux of many real-world applications of operations research, artificial intelligence, bioinformatics, etc. Despite its importance as a combinatorial substrate, many attempts for building an efficient WCSP solver have been largely unsatisfactory. In this paper, we introduce a new method for encoding a (Boolean) WCSP instance as an integer linear program (ILP). This encoding is based on the idea of the constraint composite graph (CCG) associated with a WCSP instance. We show that our CCG-based ILP encoding of the Boolean WCSP is significantly more efficient than previously known ILP encodings. Theoretically, we show that the CCG-based ILP encoding has a number of interesting properties. Empirically, we show that it allows us to solve many hard Boolean WCSP instances that cannot be solved by ILP solvers with previously known ILP encodings.
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- 1.
A MAP problem instance on a probabilistic graphic model, such as a Belief Network, can be formulated as a WCSP instance by taking the negative logarithm on the individual probabilities.
- 2.
As shown in [8], our techniques can also be generalized to the WCSP with variables of domain sizes larger than 2. However, for a proof of concept, this paper focuses on the Boolean WCSP.
- 3.
References
Berkelaar, M., Eikland, K., Notebaert, P.: lp_solve 5.5 open source (mixed integer) linear programming software (2004). http://lpsolve.sourceforge.net/5.5/
Bistarelli, S., Montanari, U., Rossi, F., Schiex, T., Verfaillie, G., Fargier, H.: Semiring-based CSPs and valued CSPs: frameworks, properties, and comparison. Constraints 4(3), 199–240 (1999)
Gurobi Optimization Inc.: Gurobi optimizer reference manual (2017) http://www.gurobi.com
Hurley, B., O’Sullivan, B., Allouche, D., Katsirelos, G., Schiex, T., Zytnicki, M., de Givry, S.: Multi-language evaluation of exact solvers in graphical model discrete optimization. Constraints 21(3), 413–434 (2016)
Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT Press, Cambridge (2009)
Kumar, T.K.S.: Incremental computation of resource-envelopes in producer-consumer models. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 664–678. Springer, Heidelberg (2003). doi:10.1007/978-3-540-45193-8_45
Kumar, T.K.S.: A framework for hybrid tractability results in Boolean weighted constraint satisfaction problems. In: Stuckey, P.J. (ed.) CP 2008. LNCS, vol. 5202, pp. 282–297. Springer, Heidelberg (2008). doi:10.1007/978-3-540-85958-1_19
Kumar, T.K.S.: Lifting techniques for weighted constraint satisfaction problems. In: The International Symposium on Artificial Intelligence and Mathematics (2008)
Kumar, T.K.S.: Kernelization, generation of bounds, and the scope of incremental computation for weighted constraint satisfaction problems. In: The International Symposium on Artificial Intelligence and Mathematics (2016)
Marinescu, R., Dechter, R.: Best-first AND/OR search for graphical models. In: The AAAI Conference on Artificial Intelligence, pp. 1171–1176 (2007)
Siek, J., Lee, L.Q., Lumsdain, A.: The Boost Graph Library: User Guide and Reference Manual. Addison-Wesley, Boston (2002)
Sierksma, G.: Linear and Integer Programming: Theory and Practice, 2nd edn. CRC Press, Boca Raton (2001)
Xu, H., Kumar, T.K.S., Koenig, S.: A new solver for the minimum weighted vertex cover problem. In: Quimper, C.-G. (ed.) CPAIOR 2016. LNCS, vol. 9676, pp. 392–405. Springer, Cham (2016). doi:10.1007/978-3-319-33954-2_28
Xu, H., Kumar, T.K.S., Koenig, S.: The Nemhauser-Trotter reduction and lifted message passing for the weighted CSP. In: Salvagnin, D., Lombardi, M. (eds.) CPAIOR 2017. LNCS, vol. 10335, pp. 387–402. Springer, Cham (2017). doi:10.1007/978-3-319-59776-8_31
Acknowledgment
The research at the University of Southern California was supported by the National Science Foundation (NSF) under grant numbers 1409987 and 1319966. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the sponsoring organizations, agencies or the U.S. government.
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Xu, H., Koenig, S., Kumar, T.K.S. (2017). A Constraint Composite Graph-Based ILP Encoding of the Boolean Weighted CSP. In: Beck, J. (eds) Principles and Practice of Constraint Programming. CP 2017. Lecture Notes in Computer Science(), vol 10416. Springer, Cham. https://doi.org/10.1007/978-3-319-66158-2_40
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DOI: https://doi.org/10.1007/978-3-319-66158-2_40
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