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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.

Keywords

Feasible Solution Constraint Programming Constraint Matrix Binary Decision Diagram Partial Assignment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Akers, S.B.: Binary decision diagrams. IEEE Transactions on Computers C-27, 509–516 (1978)CrossRefzbMATHGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    Behle, M., Eisenbrand, F.: 0/1 vertex and facet enumeration with BDDs. In: Proceedings of ALENEX. SIAM, Philadelphia (2007)Google Scholar
  6. 6.
    Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers C-35, 677–691 (1986)CrossRefzbMATHGoogle Scholar
  7. 7.
    Campos, V., Piñana, E., Martí, R.: Adaptive memory programming for matrix bandwidth minimization. Annals of Operations Research (to appear)Google Scholar
  8. 8.
    Del Corso, G.M., Manzini, G.: Finding exact solutions to the bandwidth minimization problem. Computing 62(3), 189–203 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Feige, U.: Approximating the bandwidth via volume respecting embeddings. J. Comput. Syst. Sci. 60(3), 510–539 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Hadzic, T., Hooker, J.N.: Cost-bounded binary decision diagrams for 0-1 programming. Technical report, Carnegie Mellon University (2007)Google Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Hooker, J.N.: Integrated Methods for Optimization. Springer, Heidelberg (2007)zbMATHGoogle Scholar
  17. 17.
    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)Google Scholar
  18. 18.
    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)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lee, C.Y.: Representation of switching circuits by binary-decision programs. Bell Systems Technical Journal 38, 985–999 (1959)MathSciNetCrossRefGoogle Scholar
  20. 20.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Saxe, J.: Dynamic programming algorithms for recognizing small-bandwidth graphs in polynomial time. SIAM J. Algebraic Discrete Meth. 1, 363–369 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • David Bergman
    • 1
  • Willem-Jan van Hoeve
    • 1
  • John N. Hooker
    • 1
  1. 1.Tepper School of BusinessCarnegie Mellon UniversityPittsburghU.S.A.

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