Abstract
We study the application of limited-width MDDs (multi-valued decision diagrams) as discrete relaxations for combinatorial optimization problems. These relaxations are used for the purpose of generating lower bounds. We introduce a new compilation method for constructing such MDDs, as well as algorithms that manipulate the MDDs to obtain stronger relaxations and hence provide stronger lower bounds. We apply our methodology to set covering problems, and evaluate the strength of MDD relaxations to relaxations based on linear programming. Our experimental results indicate that the MDD relaxation is particularly effective on structured problems, being able to outperform state-of-the-art integer programming technology by several orders of magnitude.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Akers, S.B.: Binary decision diagrams. IEEE Transactions on Computers C-27, 509–516 (1978)
Andersen, H.R., Hadzic, T., Hooker, J.N., Tiedemann, P.: A Constraint Store Based on Multivalued Decision Diagrams. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 118–132. Springer, Heidelberg (2007)
Becker, B., Behle, M., Eisenbrand, F., Wimmer, R.: BDDs in a branch and cut framework. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 452–463. Springer, Heidelberg (2005)
Behle, M.: On Threshold BDDs and the Optimal Variable Ordering Problem. In: Dress, A.W.M., Xu, Y., Zhu, B. (eds.) COCOA 2007. LNCS, vol. 4616, pp. 124–135. Springer, Heidelberg (2007)
Behle, M., Eisenbrand, F.: 0/1 vertex and facet enumeration with BDDs. In: Proceedings of ALENEX. SIAM, Philadelphia (2007)
Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers C-35, 677–691 (1986)
Campos, V., Piñana, E., Martí, R.: Adaptive memory programming for matrix bandwidth minimization. Annals of Operations Research (to appear)
Del Corso, G.M., Manzini, G.: Finding exact solutions to the bandwidth minimization problem. Computing 62(3), 189–203 (1999)
Feige, U.: Approximating the bandwidth via volume respecting embeddings. J. Comput. Syst. Sci. 60(3), 510–539 (2000)
Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)
Gurari, E.M., Sudborough, I.H.: Improved dynamic programming algorithms for bandwidth minimization and the mincut linear arrangement problem. ALGORITHMS: Journal of Algorithms 5 (1984)
Hadzic, T., Hooker, J.N.: Postoptimality analysis for integer programming using binary decision diagrams, presented at GICOLAG workshop (Global Optimization: Integrating Convexity, Optimization, Logic Programming, and Computational Algebraic Geometry), Vienna. Technical report, Carnegie Mellon University (2006)
Hadzic, T., Hooker, J.N.: Cost-bounded binary decision diagrams for 0-1 programming. Technical report, Carnegie Mellon University (2007)
Hadzic, T., Hooker, J.N., O’Sullivan, B., Tiedemann, P.: Approximate Compilation of Constraints into Multivalued Decision Diagrams. In: Stuckey, P.J. (ed.) CP 2008. LNCS, vol. 5202, pp. 448–462. Springer, Heidelberg (2008)
Hoda, S., van Hoeve, W.-J., Hooker, J.N.: A Systematic Approach to MDD-Based Constraint Programming. In: Cohen, D. (ed.) CP 2010. LNCS, vol. 6308, pp. 266–280. Springer, Heidelberg (2010)
Hooker, J.N.: Integrated Methods for Optimization. Springer, Heidelberg (2007)
Hu, A.J.: Techniques for Efficient Formal Verification Using Binary Decision Diagrams. Technical Report CS-TR-95-1561, Stanford University, Department of Computer Science (1995)
Kam, T., Villa, T., Brayton, R.K., Sangiovanni-Vincentelli, A.L.: Multi-valued decision diagrams: Theory and applications. International Journal on Multiple-Valued Logic 4, 9–62 (1998)
Lee, C.Y.: Representation of switching circuits by binary-decision programs. Bell Systems Technical Journal 38, 985–999 (1959)
Martí, R., Campos, V., Piñana, E.: A branch and bound algorithm for the matrix bandwidth minimization. European Journal of Operational Research 186(2), 513–528 (2008)
Martí, R., Laguna, M., Glover, F., Campos, V.: Reducing the bandwidth of a sparse matrix with tabu search. European Journal of Operational Research 135(2), 450–459 (2001)
Piñana, E., Plana, I., Campos, V., Martí, R.: GRASP and path relinking for the matrix bandwidth minimization. European Journal of Operational Research 153(1), 200–210 (2004)
Saxe, J.: Dynamic programming algorithms for recognizing small-bandwidth graphs in polynomial time. SIAM J. Algebraic Discrete Meth. 1, 363–369 (1980)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bergman, D., van Hoeve, WJ., Hooker, J.N. (2011). Manipulating MDD Relaxations for Combinatorial Optimization. In: Achterberg, T., Beck, J.C. (eds) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. CPAIOR 2011. Lecture Notes in Computer Science, vol 6697. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21311-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-21311-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21310-6
Online ISBN: 978-3-642-21311-3
eBook Packages: Computer ScienceComputer Science (R0)