Skip to main content
SpringerLink
Log in

Frameworks for Logically Classifying Polynomial-Time Optimisation Problems

  • Conference paper
Computer Science – Theory and Applications (CSR 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6072))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bueno, O., Manyem, P.: Polynomial-time maximisation classes: syntactic hierarchy. Fundamenta Informaticae 84(1), 111–133 (2008)

    MATH  MathSciNet  Google Scholar 

  2. Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, Cambridge (1990)

    MATH  Google Scholar 

  3. Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. In: Monographs in Mathematics. Springer, Heidelberg (1999)

    Google Scholar 

  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. 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)

    MATH  Google Scholar 

  6. Grädel, E.: Capturing complexity classes by fragments of second-order logic. Theoretical Computer Science 101(1), 35–57 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  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. Immerman, N.: Relational queries computable in polynomial time. Information and Control 68(1-3), 86–104 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  9. Immerman, N.: Descriptive Complexity. In: Graduate Texts in Computer Science. Springer, Heidelberg (1999)

    Google Scholar 

  10. Jaumard, B., Simeone, B.: On the complexity of the maximum satisfiability problem for horn formulas. Information Processing Letters 26(1), 1–4 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Johnson, D.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9(3), 256–278 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kohli, R., Krishnamurti, R., Mirchandani, P.: The minimum satisfiability problem. SIAM Journal of Discrete Mathematics 7(2), 275–283 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kolaitis, P.G., Thakur, M.N.: Logical definability of NP optimization problems. Information and Computation 115(2), 321–353 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kolaitis, P.G., Thakur, M.N.: Approximation properties of NP minimization classes. Journal of Computer and System Sciences 50(3), 391–411 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Libkin, L.: Elements of Finite Model Theory. In: Texts in Theoretical Computer Science. Springer, Heidelberg (2004)

    Google Scholar 

  16. Manyem, P.: Syntactic characterizations of polynomial time optimization classes. Chicago Journal of Theoretical Computer Science (3), 1–23 (2008)

    Google Scholar 

  17. Panconesi, A., Ranjan, D.: Quantifiers and approximation. Theoretical Computer Science 107(1), 145–163 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43(3), 425–440 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  19. Raman, V., Ravikumar, B., Srinivasa Rao, S.: A simplified NP-complete MAXSAT problem. Information Processing Letters 65(1), 163–168 (1998)

    Article  MathSciNet  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. School of Engineering and Computing Sciences, Durham University, Science Labs, South Road, Durham, DH1 3LE, U.K.

    James Gate & Iain A. Stewart

Authors
  1. James Gate
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iain A. Stewart
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Theoretical Cybernetics, Kazan State University, 420008, Kazan, Russia

    Farid Ablayev

  2. Institut für Informatik, Technische Universität München, 85748, Garching, Germany

    Ernst W. Mayr

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gate, J., Stewart, I.A. (2010). Frameworks for Logically Classifying Polynomial-Time Optimisation Problems. In: Ablayev, F., Mayr, E.W. (eds) Computer Science – Theory and Applications. CSR 2010. Lecture Notes in Computer Science, vol 6072. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13182-0_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-13182-0_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13181-3

  • Online ISBN: 978-3-642-13182-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation