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The Complexity of Constraints on Intervals and Lengths

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STACS 2002 (STACS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2285))

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Abstract

We study interval-valued constraint satisfaction problems (CSPs), in which the aim is to find an assignment of intervals to a given set of variables subject to constraints on the relative positions of intervals. One interesting question concerning such problems is to determine exactly how the complexity of an interval-valued CSP depends on the set of constraints allowed in instances. For the framework known as Allen’s interval algebra this question was completely answered earlier by the authors.

Here we extend the qualitative framework of Allen’s algebra with additional constraints on the lengths of intervals. We allow these length constraints to be expressed as Horn disjunctive linear relations, a wellknown tractable and sufficiently expressive form of constraints. We completely characterize sets of qualitative relations for which the constraint satisfaction problem augmented with arbitrary length constraints of the above form is tractable. We also show that all the remaining cases are NP-complete.

Partially supported by the UK EPSRC grant GR/R29598.

Partially supported by the Swedish Research Council (VR) under grant 221-2000- 361.

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References

  1. Montanari, U.: Networks of constraints: Fundamental properties and applications to picture processing. Information Sciences 7 (1974) 95–132

    Article  MathSciNet  Google Scholar 

  2. Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    Google Scholar 

  3. Angelsmark, O., Jonsson, P.: Some observations on durations, scheduling and Allen’s algebra. In: Proceedings of the 6th Conference on Constraint Programming (CP’00). Volume 1894 of Lecture Notes in Computer Science., Springer-Verlag (2000) 484–488

    Google Scholar 

  4. Jeavons, P.: On the algebraic structure of combinatorial problems. Theoretical Computer Science 200 (1998) 185–204

    Article  MATH  MathSciNet  Google Scholar 

  5. Schaefer, T.: The complexity of satisfiability problems. In: Proceedings of the 10th Symposium on the Theory of Computing (STOC’78), New-York, ACM Press (1978) 216–226

    Google Scholar 

  6. Hell, P., Nešetřil, J.: On the complexity of H-coloring. Journal of Combinatorial Theory, Ser. B 48(1990) 92–110

    Article  MATH  Google Scholar 

  7. Allen, J.: Maintaining knowledge about temporal intervals. Communications of the ACM 26(1983) 832–843

    Article  MATH  Google Scholar 

  8. Golumbic, M., Shamir, R.: Complexity and algorithms for reasoning about time: A graph-theoretic approach. Journal of the ACM 40(1993) 1108–1133

    Article  MATH  MathSciNet  Google Scholar 

  9. Krokhin, A., Jeavons, P., Jonsson, P.: Reasoning about temporal relations: Tractable subalgebras of Allen’s interval algebra. Technical Report RR-01-12, Oxford University Computing Laboratory (2001) Submitted for publication.

    Google Scholar 

  10. Nebel, B., Bürckert, H.J.: Reasoning about temporal relations: A maximal tractable subclass of Allen’s interval algebra. Journal of the ACM 42 (1995) 43–66

    Article  MATH  Google Scholar 

  11. Goldberg, P., Golumbic, M., Kaplan, H., Shamir, R.: Four strikes against physical mapping of DNA. Journal of Computational Biology 2 (1995) 139–152

    Article  Google Scholar 

  12. Golumbic, M., Kaplan, H., Shamir, R.: On the complexity of DNA physical mapping. Advances in Applied Mathematics 15(1994) 251–261

    Article  MATH  MathSciNet  Google Scholar 

  13. Karp, R.: Mapping the genome: some combinatorial problems arising in molecular biology. In: Proceedings of the 25th Symposium on the Theory of Computing (STOC’93), New-York, ACM Press (1993) 278–285

    Google Scholar 

  14. Hsu, W.L.: A simple test for interval graphs. In Hsu, W.L., Lee, R., eds.: Proceedings of the 18th InternationalWorkshop on Graph-Theoretic Concepts in Computer Science. Volume 657 of Lecture Notes in Computer Science., Springer (1992) 11–16

    Google Scholar 

  15. Corneil, D., Kim, H., Natarajan, S., Olariu, S., Sprague, A.: Simple linear time recognition of unit interval graphs. Information Processing Letters 55(1995) 99–104

    Article  MATH  MathSciNet  Google Scholar 

  16. Peér, I., Shamir, R.: Realizing interval graphs with size and distance constraints. SIAM Journal on Discrete Mathematics 10 (1997) 662–687

    Article  MathSciNet  Google Scholar 

  17. Gabor, C., Supowit, K., Hsu, W.: Recognizing circle graphs in polynomial time. Journal of the ACM 36 (1989) 435–473

    Article  MATH  MathSciNet  Google Scholar 

  18. Golumbic, M., Kaplan, H., Shamir, R.: Graph sandwich problems. Journal of Algorithms 19 (1995) 449–473

    Article  MATH  MathSciNet  Google Scholar 

  19. Peér, I., Shamir, R.: Satisfiability problems on intervals and unit intervals. Theoretical Computer Science 175 (1997) 349–372

    Article  MathSciNet  Google Scholar 

  20. Webber, A.: Proof of the interval satisfiability conjecture. Annals of Mathematics and Artificial Intelligence 15 (1995) 231–238

    Article  MATH  MathSciNet  Google Scholar 

  21. Drakengren, T., Jonsson, P.: Eight maximal tractable subclasses of Allen’s algebra with metric time. Journal of Artificial Intelligence Research7 (1997) 25–45

    MATH  MathSciNet  Google Scholar 

  22. Drakengren, T., Jonsson, P.: A complete classification of tractability in Allen’s algebra relative to subsets of basic relations. Artificial Intelligence 106 (1998) 205–219

    Article  MATH  MathSciNet  Google Scholar 

  23. Jonsson, P., Bäckström, C.: A unifying approach to temporal constraint reasoning. Artificial Intelligence 102 (1998) 143–155

    Article  MATH  MathSciNet  Google Scholar 

  24. Koubarakis, M.: Tractable disjunctions of linear constraints: basic results and applications to temporal reasoning. Theoretical Computer Science 266 (2001) 311–339

    Article  MATH  MathSciNet  Google Scholar 

  25. Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoretical Computer Science 10 (1980) 111–121

    Article  MATH  MathSciNet  Google Scholar 

  26. Creignou, N., Khanna, S., Sudan, M.: Complexity Classification of Boolean Constraint Satisfaction Problems. Volume 7 of SIAM Monographs on Discrete Mathematics and Applications. (2001)

    Google Scholar 

  27. Kirousis, L., Kolaitis, P.: The complexity of minimal satisfiability problems. In: Proceedings of the 18th Symposium on Theoretical Aspects of Computer Science (STACS 2001). Volume 2010 of Lecture Notes in Computer Science., Springer (2001) 407–418

    Google Scholar 

  28. Krokhin, A., Jeavons, P., Jonsson, P.: The complexity of constraints on intervals and lengths. Technical Report TR01-077, Electronic Colloquim on Computational Complexity (2001) (http://www.eccc.uni-trier.de/eccc).

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Author information

Authors and Affiliations

  1. Oxford University Computing Laboratory, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Andrei Krokhin & Peter Jeavons

  2. Department of Computer and Information Science, Linköping University, S-581 83, Linköping, Sweden

    Peter Jonsson

Authors
  1. Andrei Krokhin
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  2. Peter Jeavons
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  3. Peter Jonsson
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Editor information

Editors and Affiliations

  1. Institute for Computer Science, Free University of Berlin, Takustraße 9, 14195, Berlin, Germany

    Helmut Alt

  2. CNRS - I3S - INRIA Sophia Antipolis, 2004 Route des Lucioles, 06902, Sophia Antipolis, France

    Afonso Ferreira

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© 2002 Springer-Verlag Berlin Heidelberg

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Krokhin, A., Jeavons, P., Jonsson, P. (2002). The Complexity of Constraints on Intervals and Lengths. In: Alt, H., Ferreira, A. (eds) STACS 2002. STACS 2002. Lecture Notes in Computer Science, vol 2285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45841-7_36

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  • DOI: https://doi.org/10.1007/3-540-45841-7_36

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43283-8

  • Online ISBN: 978-3-540-45841-8

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