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Frameworks for Logically Classifying Polynomial-Time Optimisation Problems

  • James Gate
  • Iain A. Stewart
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6072)

Abstract

We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠ NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems.

Keywords

Feasible Solution Logical Framework Horn Clause Decision Version Relation Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bueno, O., Manyem, P.: Polynomial-time maximisation classes: syntactic hierarchy. Fundamenta Informaticae 84(1), 111–133 (2008)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, Cambridge (1990)zbMATHGoogle Scholar
  3. 3.
    Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. In: Monographs in Mathematics. Springer, Heidelberg (1999)Google Scholar
  4. 4.
    Fagin, R.: Generalized first-order spectra and polynomial-time recognizable sets. In: Complexity and Computation, SIAM-AMS Proceedings, vol. 7, pp. 43–73 (1974)Google Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  6. 6.
    Grädel, E.: Capturing complexity classes by fragments of second-order logic. Theoretical Computer Science 101(1), 35–57 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Grädel, E., Kolaitis, P.G., Libkin, L., Marx, M., Spencer, J., Vardi, M.Y., Venema, Y., Weinstein, S.: Finite Model Theory and Its Applications. In: Texts in Theoretical Computer Science. Springer, Heidelberg (2007)Google Scholar
  8. 8.
    Immerman, N.: Relational queries computable in polynomial time. Information and Control 68(1-3), 86–104 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Immerman, N.: Descriptive Complexity. In: Graduate Texts in Computer Science. Springer, Heidelberg (1999)Google Scholar
  10. 10.
    Jaumard, B., Simeone, B.: On the complexity of the maximum satisfiability problem for horn formulas. Information Processing Letters 26(1), 1–4 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Johnson, D.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9(3), 256–278 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kohli, R., Krishnamurti, R., Mirchandani, P.: The minimum satisfiability problem. SIAM Journal of Discrete Mathematics 7(2), 275–283 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kolaitis, P.G., Thakur, M.N.: Logical definability of NP optimization problems. Information and Computation 115(2), 321–353 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kolaitis, P.G., Thakur, M.N.: Approximation properties of NP minimization classes. Journal of Computer and System Sciences 50(3), 391–411 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Libkin, L.: Elements of Finite Model Theory. In: Texts in Theoretical Computer Science. Springer, Heidelberg (2004)Google Scholar
  16. 16.
    Manyem, P.: Syntactic characterizations of polynomial time optimization classes. Chicago Journal of Theoretical Computer Science (3), 1–23 (2008)Google Scholar
  17. 17.
    Panconesi, A., Ranjan, D.: Quantifiers and approximation. Theoretical Computer Science 107(1), 145–163 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43(3), 425–440 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Raman, V., Ravikumar, B., Srinivasa Rao, S.: A simplified NP-complete MAXSAT problem. Information Processing Letters 65(1), 163–168 (1998)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Vardi, M.Y.: The complexity of relational query languages. In: Proceedings of 14th ACM Ann. Symp. on the Theory of Computing, pp. 137–146 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • James Gate
    • 1
  • Iain A. Stewart
    • 1
  1. 1.School of Engineering and Computing SciencesDurham University, Science LabsDurhamU.K.

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