Constraints

, Volume 18, Issue 2, pp 166–201 | Cite as

Using dual presolving reductions to reformulate cumulative constraints

Article

Abstract

Dual presolving reductions are a class of reformulation techniques that remove feasible or even optimal solutions while guaranteeing that at least one optimal solution remains, as long as the original problem was feasible. Presolving and dual reductions are important components of state-of-the-art mixed-integer linear programming solvers. In this paper, we introduce them both as unified, practical concepts in constraint programming solvers. Building on the existing idea of variable locks, we formally define and justify the use of dual information for cumulative constraints during a presolving phase of a solver. In particular, variable locks are used to decompose cumulative constraints, detect irrelevant variables, and infer variable assignments and domain reductions. Since the computational complexity of propagation algorithms typically depends on the number of variables and/or domain size, such dual reductions are a source of potential computational speed-up. Through experimental evidence on resource constrained project scheduling problems, we demonstrate that the conditions for dual reductions are present in well-known benchmark instances and that a substantial proportion of them can be solved to optimality in presolving – without search. While we consider this result very promising, we do not observe significant change in overall run-time from the use of our novel dual reductions

Keywords

Dual reductions Cumulative constraints Presolving Variable locks 

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References

  1. 1.
    Achterberg, T. (2007). Constraint integer programming. PhD thesis, Technische Universität Berlin.Google Scholar
  2. 2.
    Achterberg, T. (2009). SCIP: Solving constraint integer programs. Mathematical Programming Computation, 1(1), 1–41.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Aggoun, A., Beldiceanu, N. (1993). Extending chip in order to solve complex scheduling and placement problems. Mathematical and Computer Modelling, 17(7), 57–73.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Artigues, C., Demassey, S., Néron, E., (eds.) (2008). Resource-constrained project scheduling: Models, algorithms, extensions and applications. iSTEGoogle Scholar
  5. 5.
    Baptiste, P., & Pape, C.L. (2000). Constraint propagation and decomposition techniques for highly disjunctive and highly cumulative project scheduling problems. Constraints, 5(1–2), 119–139.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Baptiste, P., Pape, C.L., Nuijten, W. (2001). Constraint-based scheduling. Kluwer Academic Publishers.Google Scholar
  7. 7.
    Baptiste, P., & Pape, C.L. (2005). Scheduling a single machine to minimize a regular objective function under setup constraints. Discrete Optimization, 2(1), 83–99.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Berthold, T., Heinz, S., Lübbecke, M.E., Möhring, R.H., Schulz, J. (2010). A constraint integer programming approach for resource-constrained project scheduling. In Lodi, A., Milano, M., Toth, P., (eds.) Integration of AI and OR techniques in constraint programming for combinatorial optimization problems (vol. 6140, pp. 313–317) of LNCS. Springer.Google Scholar
  9. 9.
    Berthold, T., Heinz, S., Schulz, J. (2011). An approximative criterion for the potential of energetic reasoning. In Marchetti-Spaccamela, A., Segal, M., (eds.) Theory and Practice of Algorithms in (Computer) systems (vol. 6595, pp. 229–239) of LNCS. Springer.Google Scholar
  10. 10.
    Bixby, R.E., & Wagner, D.K. (1987). A note on detecting simple redundancies in linear systems. Operation Research Letters, 6(1), 15–17.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bordeaux, L., Cadoli, M., Mancini, T. (2008). A unifying framework for structural properties of CSPs: definitions, complexity, tractabilit. Journal of Artificial Intelligence Research, 32(1), 607–629.MathSciNetMATHGoogle Scholar
  12. 12.
    Borrett, J.E., & Tsang, E.P.K. (2001). A context for constraint satisfaction problem formulation selection. Constraints, 6(4), 299–327.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Chu, G., & Stuckey, P.J. (2012). A generic method for identifying and exploiting dominance relations. In Milano, M., (ed.) Principles and practice of constraint programming - CP 2012 (vol. 7514, pp. 6–22) of LNCS.Google Scholar
  14. 14.
    Dantzig, G.B., & Thapa, M.N. (2003). Linear programming 2. Springer Series in Operations Research. Springer.Google Scholar
  15. 15.
    de la Banda, M.J.G., Marriott, K., Rafeh, R., Wallace, M. (2006). The modelling language Zinc. In Benhamou, F., (ed.) Principles and practice of constraint programming - CP 2006 (vol. 4204, pp. 700–705). LNCS.Google Scholar
  16. 16.
    Dechter, R. (2003). Constraint processing. Elsevier Morgan Kaufmann.Google Scholar
  17. 17.
    Frisch, A.M., Harvey, W., Jefferson, C., Martínez-Hernández, B., Miguel, I. (2008). ESSENCE: A constraint language for specifying combinatorial problems. Constraints, 13, 268–306.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Gent, I.P., Petrie, K.E., Puget, J.F. (2006). Symmetry in constraint programming. In Handbooks of constraint programming. Elsevier.Google Scholar
  19. 19.
    Gent, I.P., Miguel, I., Rendl, A. (2007). Tailoring solver-independent constraint models: a case study with ESSENCE and MINION. In Miguel, I., Ruml, W., (eds.) Abstraction, Reformulation, and Approximation (vol. 4612, pp. 184–199) of LNCS.Google Scholar
  20. 20.
    Guzelsoy, M. (2010). Dual methods in mixed integer linear programming. PhD thesis, Lehigh University,Industrial and Systems Engineering.Google Scholar
  21. 21.
    Heinz, S., & Schulz, J. (2011). Explanations for the cumulative constraint: An experimental study. In Pardalos, P.M., Rebennack, S. (eds.) Experimental algorithms (vol. 6630, pp. 400–409) of LNCS. Springer.Google Scholar
  22. 22.
    Imbert, J.L., & Hentenryck, P.V. (1996). Redundancy elimination with a lexicographic solved form. Annals of Mathematics and Artificial Intelligence, 17(1–2), 85–106.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Karwan, M.H., Lotfi, V., Telgen, J., Zionts, S. (1983). Redundancy in mathematical programming. A state-of-the-art survey. Volume 206 of Lecture Notes in Economics and Mathematical Systems. Springer.Google Scholar
  24. 24.
    Kiziltan, Z. (2004). Symmetry breaking ordering constraints: Thesis. AI Communications, 17, 167–169.MathSciNetGoogle Scholar
  25. 25.
    Klein, R., & Scholl, A. (1999). Computing lower bounds by destructive improvement: An application to resource-constrained project scheduling. European Journal of Operational Research, 112(2), 322–346.MATHCrossRefGoogle Scholar
  26. 26.
    Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K. (2011). MIPLIB 2010. Mathematical Programming Computation, 3(2), 103–163.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Laurière, J.L. (1978). A language and a program for stating and solving combinatorial problems. Artificial Intelligence, 10(1), 29–127.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Mahajan, A. (2010). Presolving mixed-integer linear programs. Preprint ANL/MCS-P1752-0510, Mathematics and Computer Science Division.Google Scholar
  29. 29.
    Marriott, K., Nethercote, N., Rafeh, R., Stuckey, P.J., de la Banda, M.G., Wallace, M. (2008). The design of the zinc modelling language. Constraints, 13(3), 229–267.MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Möhring, R.H., Schulz, A.S., Stork, F., Uetz, M. (2003). Solving project scheduling problems by minimum cut computations. Management Science, 49(3), 330–350.MATHCrossRefGoogle Scholar
  31. 31.
    Nethercote, N., Stuckey, P.J., Becket, R., Brand, S., Duck, G.J., Tack, G. (2007). MiniZinc: Towards a standard CP modelling language. In Bessiere, C., (ed.) Principles and Practice of Constraint Programming - CP 2007 (vol. 4741, pp. 529–543) of LNCS. Springer.Google Scholar
  32. 32.
    Pesant, G., Quimper, C., Zanarini, A. (2012). Counting-based search: Branching heuristics for constraint satisfaction problems. Journal of Artificial Intelligence Research, 43, 173–210.MathSciNetMATHGoogle Scholar
  33. 33.
    Prestwich, S.D., & Beck, J.C. (2004). Exploiting dominance in three symmetric problems. In Proceedings of the fourth international workshop on symmetry and constraint satisfaction problems.Google Scholar
  34. 34.
    PSPLib: Project scheduling problem library. http://129.187.106.231/psplib/.
  35. 35.
    Puget, J.F. (2005). Automatic detection of variable and value symmetries. In van Beek, P. (ed.) Principles and practice of constraint programming - CP 2005 (vol. 3709, pp. 475–489) of LNCS.Google Scholar
  36. 36.
    Rendl, A. (2010). Effective compilation of constraint models. PhD thesis, University of St Andrews.Google Scholar
  37. 37.
    Savelsbergh, M.W.P. (1994). Preprocessing and probing techniques for mixed integer programming problems. ORSA Journal on Computing, 6, 445–454.MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Schutt, A., Feydy, T., Stuckey, P.J., Wallace, M.G. (2012). Solving rcpsp/max by lazy clause generation. Journal of Scheduling. (accepted).Google Scholar
  39. 39.
    van Hoeve, W.J. (2001). The all different constraint: A survey. CoRR cs.PL/0105015.Google Scholar
  40. 40.
    Vilím, P. (2009). Max energy filtering algorithm for discrete cumulative resources. In van Hoeve, W.J., Hooker, J.N. (eds.) Integration of AI and OR techniques in constraint programming for combinatorial optimization problems (vol. 5547, pp. 294–308) of LNCS.Google Scholar
  41. 41.
    Yunes, T., Aron, I.D., Hooker, J.N. (2010). An integrated solver for optimization problems. Operations Research, 58(2), 342–356.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Stefan Heinz
    • 1
  • Jens Schulz
    • 2
  • J. Christopher Beck
    • 3
  1. 1.Zuse Institute BerlinBerlinGermany
  2. 2.Technische Universität BerlinInstitut für MathematikBerlinGermany
  3. 3.Department of Mechanical & Industrial EngineeringUniversity of TorontoTorontoCanada

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