On-line algorithms for networks of temporal constraints

  • Fabrizio d'Amore
  • Fabio Iacobini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1335)


We consider a semi-dynamic setting for the Temporal Constraint Satisfaction Problem, where we are requested to maintain the path-consistency of a network under a sequence of insertions of new (further) constraints between pairs of variables. We show how to maintain path-consistent a network in the defined setting in O(nR3) amortized time on a sequence of Θ(n2) insertions, where n is the number of vertices of the network and R is its range, defined as the maximum size of the minimum interval containing all the intervals of a single constraint. Furthermore we extend our algorithms to deal with more general temporal networks where variables can be points and/or intervals and constraints can be also defined on pairs of variables of different kind. For such cases our algorithms maintain their performance. Finally we adapt our algorithms for maintaining also the arc-consistency of such general networks, which is a particular kind of path-consistency limited to paths of length 1. The property is maintained in O(R) amortized time for Θ(n2) insertions. In case of constraints consisting of simple intervals the algorithm also gives a solution to the satisfaction problem.


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  1. 1.
    J.F. Allen. Maintaining knowledge about temporal intervals. Communication of the ACM, 26(11), 1983.Google Scholar
  2. 2.
    F.D. Anger and R.V. Rodriguez. Effective scheduling of tasks under weak temporal interval constraints. Lecture Notes in Computer Science, 945, 1995.Google Scholar
  3. 3.
    J. Carmo and A. Sernadas. A temporal logic framework for a layered approach to systems specification and verification. In Proceedings of the Conference on Temporal Aspects in Information Systems, pages 31–47, France, May 1987. AFCET.Google Scholar
  4. 4.
    R. Cervoni, A. Cesta, and A. Oddi. Managing dynamic temporal constraint networks. In Proceedings of AIPS '94, 1994.Google Scholar
  5. 5.
    E. Davis, 1989. Private communication reported in [6].Google Scholar
  6. 6.
    R. Dechter, I. Meiri, and J. Pearl. Temporal constraint networks. Artificial Intelligence, 49, 1991.Google Scholar
  7. 7.
    E.C. Freuder. A sufficient condition for backtrack-free search. Journal of the ACM, 29, 1982.Google Scholar
  8. 8.
    C.-C. Han and C.H. Lee. Comments on Mohr and Hendersons path consistency algorithms. Artificial Intelligence, 36, 1988.Google Scholar
  9. 9.
    C.-C. Han, K.-J. Lin, and J.W.-S. Liu. Scheduling jobs with temporal distance constraints. SIAM Journal on Computing, 24(5):1104–1121, 1995.Google Scholar
  10. 10.
    A.K. Mackworth. Consistency in networks of relations. Artificial Intelligence, 8, 1977.Google Scholar
  11. 11.
    Z. Manna and A. Pnueli. Verification of concurrent programs: Temporal proof principle. In D. Kozen, editor, Logics of Programs (Proceedings 1981), LNCS 131, pages 200–252. Springer-Verlag, 1981.Google Scholar
  12. 12.
    Z. Manna and A. Pnueli. Verification of concurrent programs: the temporal framework. In R.S. Boyer and J.S. Moore, editors, The Correctness Problem in Computer Science, pages 215–273. Academic Press, 1981.Google Scholar
  13. 13.
    I. Meiri. Combining qualitative and quantitative constraints in temporal reasoning. In Proc. of the 10th National Conference of the American Association for Artificial Intelligence (AAAI '91), 1991.Google Scholar
  14. 14.
    U. Montanari. Networks of constraints: Fundamental properties and applications to picture processing. Information Sciences, 7, 1974.Google Scholar
  15. 15.
    B. Nebel and H.J. Biirckert. Reasoning about temporal relations: a maximal tractable subclass of Allen's interval algebra. Journal of the ACM, 42, 1995.Google Scholar
  16. 16.
    H. Noltemeier and G. Schmitt. Incremental temporal constraint propagation. In Proc. of the 9th Florida Artificial Intelligence Research Symp. (FLAIRS '96), pages 25–29, 1996.Google Scholar
  17. 17.
    J.S. Ostroff. Temporal Logic of Real-Time Systems. Research Studies Press, 1990.Google Scholar
  18. 18.
    A. Tansel, J. Clifford, S. Gadia, S. Jajodia, A. Segev, and R. Snodgrass, editors. Temporal Databases: Theory, Design, and Implementation. Database Systems and Applications Series. Benjamin/Cummings, Redwood City, CA, 1993.Google Scholar
  19. 19.
    R.E. Tarjan. Amortized computational complexity. SIAM J. Algebraic Discrete Methods, 6(2):306–318, 1985.Google Scholar
  20. 20.
    P. van Beek. Reasoning about qualitative temporal information. Artificial intelligence, 58, 1992.Google Scholar
  21. 21.
    M. Vilain and H.A. Kautz. Constraint propagation algorithms for temporal reasoning. In Proc. of the 5th National Conference of the American Association for Artificial Intelligence (AAAI '86), 1986.Google Scholar
  22. 22.
    M. Vilain, H.A. Kautz, and P. van Beek. Constraint propagation algorithms for temporal reasoning: a revised report. In D.S. Weld and J. de Kleer, editors, Readings in qualitative reasoning about physical systems. Morgan Kaufman, 1989.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Fabrizio d'Amore
    • 1
  • Fabio Iacobini
    • 1
  1. 1.Dipartimento di Informativa e SistemisticaUniversità di Roma “La Sapienza”RomaItaly

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