Abstract
Inter-block backtracking (IBB) computes all the solutions of sparse systems of non-linear equations over the reals. This algorithm, introduced in 1998 by Bliek et al., handles a system of equations previously decomposed into a set of (small) k ×k sub-systems, called blocks. Partial solutions are computed in the different blocks and combined together to obtain the set of global solutions.
When solutions inside blocks are computed with interval-based techniques, IBB can be viewed as a new interval-based algorithm for solving decomposed equation systems. Previous implementations used Ilog Solver and its IlcInterval library. The fact that this interval-based solver was more or less a black box implied several strong limitations.
The new results described in this paper come from the integration of IBB with the interval-based library developed by the second author. This new library allows IBB to become reliable (no solution is lost) while still gaining several orders of magnitude w.r.t. solving the whole system. We compare several variants of IBB on a sample of benchmarks, which allows us to better understand the behavior of IBB. The main conclusion is that the use of an interval Newton operator inside blocks has the most positive impact on the robustness and performance of IBB. This modifies the influence of other features, such as intelligent backtracking and filtering strategies.
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References
Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.-F.: Revising Hull and Box Consistency. In: ICLP, pp. 230–244 (1999)
Bliek, C., Neveu, B., Trombettoni, G.: Using Graph Decomposition for Solving Continuous CSPs. In: Maher, M.J., Puget, J.-F. (eds.) CP 1998. LNCS, vol. 1520, pp. 102–116. Springer, Heidelberg (1998)
Bouma, W., Fudos, I., Hoffmann, C.M., Cai, J., Paige, R.: Geometric constraint solver. Computer Aided Design 27(6), 487–501 (1995)
Chabert, G.: Contributions á la résolution de Contraintes sur intervalles? Ph.D thesis, Université de Nice–Sophia Antipolis (2006) (to be defended)
Debruyne, R., Bessière, C.: Some Practicable Filtering Techniques for the Constraint Satisfaction Problem. In: Proc. of IJCAI, pp. 412–417 (1997)
Dechter, R.: Enhancement schemes for constraint processing: Backjumping, learning, and cutset decomposition. Artificial Intelligence 41(3), 273–312 (1990)
Granvilliers, L.: RealPaver User’s Manual, version 0.3. University of Nantes (2003), Available at: www.sciences.univ-nantes.fr/info/perso/permanents/granvil/realpaver
Granvilliers, L.: Realpaver: An interval solver using constraint satisfaction techniques. ACM Transactions on Mathematical Software (accepted for publication)
Van Hentenryck, P., Michel, L., Deville, Y.: Numerica: A Modeling Language for Global Optimization. MIT Press, Cambridge (1997)
Hoffmann, C., Lomonossov, A., Sitharam, M.: Finding solvable subsets of constraint graphs. In: Smolka, G. (ed.) CP 1997. LNCS, vol. 1330, pp. 463–477. Springer, Heidelberg (1997)
ILOG, Av. Galliéni, Gentilly. Ilog Solver V. 5, Users’ Reference Manual (2000)
Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis. Springer, Heidelberg (2001)
Jermann, C., Neveu, B., Trombettoni, G.: Algorithms for Identifying Rigid Subsystems in Geometric Constraint Systems. In: Proc. IJCAI, pp. 233–238 (2003)
Neveu, B., Jermann, C., Trombettoni, G.: Inter-block Backtracking: Exploiting the Structure in Continuous CSPs. In: Jermann, C., Neumaier, A., Sam, D. (eds.) COCOS 2003. LNCS, vol. 3478, pp. 15–30. Springer, Heidelberg (2005)
Jermann, C., Trombettoni, G., Neveu, B., Mathis, P.: Decomposition of Geometric Constraint Systems: a Survey. Int. Journal of Computational Geometry and Applications (IJCGA) 16 (2006)
Latham, R.S., Middleditch, A.E.: Connectivity analysis: A tool for processing geometric constraints. Computer Aided Design 28(11), 917–928 (1996)
Lebbah, Y.: Contribution á la résolution de Contraintes par Consistance Forte. Ph.D thesis, Université de Nantes (1999)
Lebbah, Y., Michel, C., Rueher, M., Daney, D., Merlet, J.P.: Efficient and safe global constraints for handling numerical constraint systems. SIAM Journal on Numerical Analysis 42(5), 2076–2097 (2005)
Lhomme, O.: Consistency Tech. for Numeric CSPs. In: IJCAI, pp. 232–238 (1993)
McAllester, D.A.: Partial order backtracking. Research Note, Artificial Intelligence Laboratory. MIT (1993), ftp://ftp.ai.mit.edu/people/dam/dynamic.ps
Merlet, J.-P.: Optimal design for the micro robot. In: IEEE Int. Conf. on Robotics and Automation (2002)
Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)
Wilczkowiak, M., Trombettoni, G., Jermann, C., Sturm, P., Boyer, E.: Scene Modeling Based on Constraint System Decomposition Tech. In: Proc. ICCV (2003)
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Neveu, B., Chabert, G., Trombettoni, G. (2006). When Interval Analysis Helps Inter-block Backtracking. In: Benhamou, F. (eds) Principles and Practice of Constraint Programming - CP 2006. CP 2006. Lecture Notes in Computer Science, vol 4204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11889205_29
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DOI: https://doi.org/10.1007/11889205_29
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