Skip to main content

Optimization Problems, Approximation Schemes, and Their Relation to FPT

  • Chapter
Fundamentals of Parameterized Complexity

Part of the book series: Texts in Computer Science ((TCS))

  • 3361 Accesses

Abstract

We begin to analyze the relationship between classical optimization problems and parameterized complexity. We explore syntactically defined classes of optimization problems that capture some of the main issues.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover 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

Notes

  1. 1.

    The appropriate reduction here is called an L-reduction. See Exercise 9.5.4.

References

  1. S. Arora, C. Lund, R. Motwani, M. Sudan, M. Szegedy, Proof verification and hardness of approximation problems, in Proceedings of 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992, Pittsburgh, Pennsylvania, USA, 24–27 October 1992 (IEEE Comput. Soc., Los Alamitos, 1992), pp. 14–23

    Google Scholar 

  2. C. Bazgan, Schémas d’approximation et complexité paramétrée, Rapport de stage de DEA d’Informatique á Orsay, Université Paris-Sud, 1995

    Google Scholar 

  3. G. Brightwell, P. Winkler, Counting linear extensions. Order 8(3), 225–242 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Brightwell, P. Winkler, Counting linear extensions is #P-complete, in Proceedings of 23rd ACM Symposium on Theory of Computing (STOC ’91), New Orleans, Louisiana, USA, May 6–May 8, 1991, ed. by C. Koutsougeras, J.S. Vitter (ACM, New York, 1991), pp. 175–181

    Google Scholar 

  5. L. Cai, Fixed parameter tractability and approximation problems, Project report, June 1992

    Google Scholar 

  6. L. Cai, J. Chen, On fixed-parameter tractability and approximability of NP optimization problems. J. Comput. Syst. Sci. 54(3), 465–474 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Chen, J. Flum, M. Grohe, An analysis of the W ∗-hierarchy. J. Symb. Log. 72, 513–534 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Downey, M. Fellows, K. Regan, Descriptive complexity and the W-hierarchy, in DIMACS Workshop on Proof Complexity and Feasible Arithmetics, April 1996, ed. by P. Beame, S. Buss. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 39 (Am. Math. Soc., Providence, 1997), pp. 119–134

    Google Scholar 

  9. H.-D. Ebbinghaus, J. Flum, Finite Model Theory (Springer, Berlin, 1995)

    Book  MATH  Google Scholar 

  10. R. Fagin, Generalized first order spectra and polynomial-time recognizable sets, in Complexity of Computation, ed. by R. Karp. SIAM–AMS Proceedings, vol. 7 (Am. Math. Soc., Providence, 1974), pp. 43–73

    Google Scholar 

  11. R. Fagin, Finite model theory—a personal perspective. Theor. Comput. Sci. 116(1), 3–31 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Flum, M. Grohe, Fixed-parameter tractability, definability, and model checking. SIAM J. Comput. 31(1), 113–145 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Flum, M. Grohe, The parameterized complexity of counting problems, in Proceedings of 43rd Symposium on Foundations of Computer Science, FOCS 2002, Vancouver, BC, Canada, 16–19 November 2002 (IEEE Comput. Soc., Los Alamitos, 2002), pp. 538–547

    Google Scholar 

  14. J. Flum, M. Grohe, Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series (Springer, Berlin, 2006)

    Google Scholar 

  15. M. Garey, D. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979)

    MATH  Google Scholar 

  16. O. Ibarra, C. Kim, Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22(4), 463–468 (1975)

    MathSciNet  MATH  Google Scholar 

  17. N. Immermann, Expressibility as a complexity measure: results and directions, in Proceedings of Second Annual Structure in Complexity Theory Conference, Cornell University, Ithaca, NY, June 16–19, 1987 (IEEE Comput. Soc., Los Alamitos, 1987), pp. 194–202

    Google Scholar 

  18. D. Johnson, A. Demers, J. Ullman, M. Garey, R. Graham, Worst case performance bounds for simple one dimensional packing algorithms. SIAM J. Comput. 3(4), 299–325 (1974)

    Article  MathSciNet  Google Scholar 

  19. H. Kierstead, W. Trotter, An extremal problem in recursive combinatorics. Congr. Numer. 33, 143–153 (1981)

    MathSciNet  Google Scholar 

  20. E. Leggett, D. Moore, Optimization problems and the polynomial time hierarchy. Theor. Comput. Sci. 15, 279–289 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. L. Libkin, Elements of Finite Model Theory. Texts in Theoretical Computer Science. An EATCS Series (Springer, Berlin, 2004)

    Book  MATH  Google Scholar 

  22. R. Lipton, R. Tarjan, Applications of a planar separator theorem, in Proceedings of 18th Annual Symposium on Foundations of Computer Science, FOCS 1977, Providence, Rhode Island, USA, 31 October–1 November 1977 (IEEE Comput. Soc., Los Alamitos, 1977), pp. 162–170

    Google Scholar 

  23. C. Papadimitriou, M. Yannakakis, Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 425–440 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag London

About this chapter

Cite this chapter

Downey, R.G., Fellows, M.R. (2013). Optimization Problems, Approximation Schemes, and Their Relation to FPT . In: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-5559-1_9

Download citation

  • DOI: https://doi.org/10.1007/978-1-4471-5559-1_9

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-5558-4

  • Online ISBN: 978-1-4471-5559-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics