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Job Sequencing Bounds from Decision Diagrams

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Book cover Principles and Practice of Constraint Programming (CP 2017)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 10416))

Abstract

In recent research, decision diagrams have proved useful for the solution of discrete optimization problems. Their success relies on the use of relaxed decision diagrams to obtain bounds on the optimal value, either through a node merger or a node splitting mechanism. We investigate the potential of node merger to provide bounds for dynamic programming models that do not otherwise have a practical relaxation, in particular the job sequencing problem with time windows and state-dependent processing times. We prove general conditions under which a node merger operation yields a valid relaxation and apply them to job sequencing. Computational experiments show that, surprisingly, relaxed diagrams prove the optimal value when their size is only a small fraction of the size of an exact diagram. On the other hand, a relaxed diagram of fixed size ceases to provide a useful bound as the instances scale up.

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Notes

  1. 1.

    Problems with nonseparable objective functions can also be represented, as described in [21], but to simplify exposition we omit this possibility.

References

  1. Akers, S.B.: Binary decision diagrams. IEEE Trans. Comput. C–27, 509–516 (1978)

    Article  MATH  Google Scholar 

  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). doi:10.1007/978-3-540-74970-7_11

    Chapter  Google Scholar 

  3. Baldacci, R., Mingozzi, A., Roberti, R.: New state-space relaxations for solving the traveling salesman problem with time windows. INFORMS J. Comput. 24(3), 356–371 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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). doi:10.1007/11427186_39

    Chapter  Google Scholar 

  5. Behle, M., Eisenbrand, F.: 0/1 vertex and facet enumeration with BDDs. In: Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 158–165. SIAM (2007)

    Google Scholar 

  6. Bergman, D., Ciré, A.A., van Hoeve, W.-J., Hooker, J.N.: Variable ordering for the application of BDDs to the maximum independent set problem. In: Beldiceanu, N., Jussien, N., Pinson, É. (eds.) CPAIOR 2012. LNCS, vol. 7298, pp. 34–49. Springer, Heidelberg (2012). doi:10.1007/978-3-642-29828-8_3

    Chapter  Google Scholar 

  7. Bergman, D., Ciré, A.A., van Hoeve, W.J., Hooker, J.N.: Optimization bounds from binary decision diagrams. INFORMS J. Comput. 26, 253–268 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bergman, D., Ciré, A.A., van Hoeve, W.J., Hooker, J.N.: Decision Diagrams for Optimization. Springer, Heidelberg (2016). doi:10.1007/978-3-319-42849-9

    Book  MATH  Google Scholar 

  9. Bergman, D., Ciré, A.A., van Hoeve, W.J., Hooker, J.N.: Discrete optimization with binary decision diagrams. INFORMS J. Comput. 28, 47–66 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bergman, D., Ciré, A.A., van Hoeve, W.J.: Lagrangian bounds from decision diagrams. Constraints 20, 346–361 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bergman, D., van Hoeve, W.-J., Hooker, J.N.: Manipulating MDD relaxations for combinatorial optimization. In: Achterberg, T., Beck, J.C. (eds.) CPAIOR 2011. LNCS, vol. 6697, pp. 20–35. Springer, Heidelberg (2011). doi:10.1007/978-3-642-21311-3_5

    Chapter  Google Scholar 

  12. Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Comput. C–35, 677–691 (1986)

    Article  MATH  Google Scholar 

  13. Christofides, N., Mingozzi, A., Toth, P.: State-space relaxation procedures for the computation of bounds to routing problems. Networks 11(2), 145–164 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ciré, A.A., van Hoeve, W.J.: Multivalued decision diagrams for sequencing problems. Oper. Res. 61, 1411–1428 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hadžić, T., Hooker, J.N.: Postoptimality analysis for integer programming using binary decision diagrams. Carnegie Mellon University, Technical report (2006)

    Google Scholar 

  16. Hadžić, T., Hooker, J.N.: Cost-bounded binary decision diagrams for 0-1 programming. In: Hentenryck, P., Wolsey, L. (eds.) CPAIOR 2007. LNCS, vol. 4510, pp. 84–98. Springer, Heidelberg (2007). doi:10.1007/978-3-540-72397-4_7

    Chapter  Google Scholar 

  17. Hadžić, 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). doi:10.1007/978-3-540-85958-1_30

    Chapter  Google Scholar 

  18. Hadžić, T., Hooker, J.N., Tiedemann, P.: Propagating separable equalities in an MDD store. In: Perron, L., Trick, M.A. (eds.) CPAIOR 2008. LNCS, vol. 5015, pp. 318–322. Springer, Heidelberg (2008). doi:10.1007/978-3-540-68155-7_30

    Chapter  Google Scholar 

  19. 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). doi:10.1007/978-3-642-15396-9_23

    Chapter  Google Scholar 

  20. Hooker, J.N.: Discrete global optimization with binary decision diagrams. In: GICOLAG 2006, Vienna, Austria, December 2006

    Google Scholar 

  21. Hooker, J.N.: Decision diagrams and dynamic programming. In: Gomes, C., Sellmann, M. (eds.) CPAIOR 2013. LNCS, vol. 7874, pp. 94–110. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38171-3_7

    Chapter  Google Scholar 

  22. Hu, A.J.: Techniques for efficient formal verification using binary decision diagrams. Thesis CS-TR-95-1561, Stanford University, Department of Computer Science, December 1995

    Google Scholar 

  23. Lee, C.Y.: Representation of switching circuits by binary-decision programs. Bell Syst. Tech. J. 38, 985–999 (1959)

    Article  MathSciNet  Google Scholar 

  24. Loekito, E., Bailey, J., Pei, J.: A binary decision diagram based approach for mining frequent subsequences. Knowl. Inf. Syst. 24(2), 235–268 (2010)

    Article  Google Scholar 

  25. Mingozzi, A.: State space relaxation and search strategies in dynamic programming. In: Koenig, S., Holte, R.C. (eds.) SARA 2002. LNCS, vol. 2371, p. 51. Springer, Heidelberg (2002). doi:10.1007/3-540-45622-8_4

    Chapter  Google Scholar 

  26. Righini, G., Salani, M.: New dynamic programming algorithms for the resource constrained shortest path problem. Networks 51, 155–170 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wegener, I.: Branching Programs and Binary Decision Diagrams: Theory and Applications. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (2000)

    Google Scholar 

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Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, USA

    J. N. Hooker

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  1. J. N. Hooker
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Correspondence to J. N. Hooker .

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  1. University of Toronto, Toronto, Ontario, Canada

    J. Christopher Beck

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Hooker, J.N. (2017). Job Sequencing Bounds from Decision Diagrams. 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_36

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  • DOI: https://doi.org/10.1007/978-3-319-66158-2_36

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