Integer Programs and Valid Inequalities for Planning Problems

  • Alexander Bockmayr
  • Yannis Dimopoulos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1809)


Part of the recent work in AI planning is concerned with the development of algorithms that regard planning as a combinatorial search problem. The underlying representation language is basically propositional logic. While this is adequate for many domains, it is not clear if it remains so for problems that involve numerical constraints, or optimization of complex objective functions. Moreover, the propositional representation imposes restrictions on the domain knowledge that can be utilized by these approaches. In order to address these issues, we propose moving to the more expressive language of Integer Programming (IP). We show how capacity constraints can be easily encoded into linear 0-1 inequalities and how rich forms of domain knowledge can be compactly represented and computationally exploited by IP solvers. Then we introduce a novel heuristic search method based on the linear programming relaxation. Finally, we present the results of our experiments with a classical relaxation-based IP solver and a logic-based 0-1 optimizer.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barth, P.: Logic-based 0-1 constraint programming. Operations Research Computer Science Interfaces Series. Kluwer, Dordrecht (1996), Google Scholar
  2. 2.
    Barth, P., Bockmayr, A.: Modelling discrete optimisation problems in constraint logic programming. Annals of Operations Research 81, 467–496 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blum, A., Furst, M.: Fast planning through planning graph analysis. Artificial Intelligence 90, 281–300 (1997)MATHCrossRefGoogle Scholar
  4. 4.
    Bockmayr, A.: Solving pseudo-Boolean constraints. In: Podelski, A. (ed.) Constraint Programming: Basics and Trends. LNCS, vol. 910, pp. 22–38. Springer, Heidelberg (1995)Google Scholar
  5. 5.
    Bockmayr, A., Dimopoulos, Y.: Mixed integer programming models for planning problems. In: CP 1998 Workshop on Constraint Problem Reformulation (1998)Google Scholar
  6. 6.
    Bonet, B., Loerincs, G., Geffner, H.: A fast and robust action selection mechanism for planning. In: AAAI 1997 (1997)Google Scholar
  7. 7.
    Bylander, T.: A Linear Programming Heuristic for Optimal Planning. In: AAAI 1997 (1997)Google Scholar
  8. 8.
    CPLEX Optimization, Inc. Using the CPLEX callable library (1995),
  9. 9.
    Gerevini, A., Schubert, L.: Accelerating Partial-Order Planners: Some Techniques for Effective Search Control and Pruning. Journal of Artificial Intelligence Research 5 (1996)Google Scholar
  10. 10.
    Huang, Y.-C., Selman, B., Kautz, H.: Control Knowledge in Planning: Benefits and Tradeoffs. In: IJCAI 1999 (1999)Google Scholar
  11. 11.
    Kautz, H., Selman, B.: Pushing the envelope: planning, propositional logic and stochastic search. In: AAAI 1996 (1996)Google Scholar
  12. 12.
    Kautz, H., Selman, B.: The Role of Domain-Specific Knowledge in the Planning as Satisfiability Framework. In: AIPS 1998 (1998)Google Scholar
  13. 13.
    Kautz, H., Selman, B.: Unifying SAT-based and Graph-based Planning. In: IJCAI 1999 (1999)Google Scholar
  14. 14.
    Kautz, H., Walser, J.: State-Space Planning by Integer Optimization. In: AAAI 1999 (1999)Google Scholar
  15. 15.
    Koehler, J.: Planning under Resource Constraints. In: ECAI 1998 (1998)Google Scholar
  16. 16.
    Koehler, J., Nebel, B., Hoffmann, J., Dimopoulos, Y.: Extending Planning Graphs to an ADL Subset. In: Steel, S. (ed.) ECP 1997. LNCS, vol. 1348. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  17. 17.
    McDermott, D.: A heuristic estimator for means-ends analysis in planning. In: AIPS 1996 (1996)Google Scholar
  18. 18.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, Chichester (1988)MATHGoogle Scholar
  19. 19.
    van Beek, P., Chen, X.: CPlan: A Constraint Programming Approach to Planning. In: AAAI 1999 (1999)Google Scholar
  20. 20.
    Vossen, T., Ball, M., Lotem, A., Nau, D.: On the Use of Integer Programming Models in AI Planning. In: IJCAI 1999 (1999)Google Scholar
  21. 21.
    Wolfman, S., Weld, D.: The LPSAT Engine and its Application to Resource Planning. In: IJCAI 1999 (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Alexander Bockmayr
    • 1
  • Yannis Dimopoulos
    • 2
  1. 1.Université Henri Poincaré, LORIAVandœvre-lès-NancyFrance
  2. 2.Dep. of Computer ScienceUniversity of CyprusNicosiaCyprus

Personalised recommendations