Skip to main content
SpringerLink
Log in

Fixed-Parameter Approximation: Conceptual Framework and Approximability Results

  • Conference paper
Parameterized and Exact Computation (IWPEC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4169))

Included in the following conference series:

Abstract

The notion of fixed-parameter approximation is introduced to investigate the approximability of optimization problems within the framework of fixed-parameter computation. This work partially aims at enhancing the world of fixed-parameter computation in parallel with the conventional theory of computation that includes both exact and approximate computations. In particular, it is proved that fixed-parameter approximability is closely related to the approximation of small-cost solutions in polynomial time. It is also demonstrated that many fixed-parameter intractable problems are not fixed-parameter approximable. On the other hand, fixed-parameter approximation appears to be a viable approach to solving some inapproximable yet important optimization problems. For instance, all problems in the class MAX SNP admit fixed-parameter approximation schemes in time O(2\(^{O((1-{\epsilon}/{\it O}(1)){\it k})}\) p(n)) for any small ε> 0.

The authors would like to thank Rod Downey and Mike Fellows regarding the definition of the fixed-parameter approximability for the problem dominating set.

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. Alber, J., Fernau, H., Niedermeier, R.: Parameterized complexity: exponential speed-up for planar graph problems. J. Algorithms 52, 26–56 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and their Approximability Properties. Springer, Heidelberg (1999)

    MATH  Google Scholar 

  3. Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41, 153–180 (1994)

    Article  MATH  Google Scholar 

  4. Bodlaender, H.: (unpublished manuscipt)

    Google Scholar 

  5. Bodlaender, H.L.: A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth. SIAM J. Comput. 25, 1305–1317 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bodlaender, H., Fellows, M.R.: (unpublished manuscipt)

    Google Scholar 

  7. Cai, L., Chen, J.: On fixed-parameter tractability and approximability of NP optimization problems. J. of Comp. and Sys. Sciences 54, 465–474 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cai, L., Chen, J.: On the amount of nondeterminism and the power of verifying. SIAM Journal of Computing 26, 733–750 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cai, L., Chen, J., Downey, R., Fellows, M.: On the structure of parameterized problems in NP. Information and Computation 123, 38–49 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cai, L., Juedes, D.W.: On the existence of sub-exponential time parameterized algorithms. Journal of Computer and System Sciences 67, 789–807 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cai, L., Juedes, D., Kanj, I.: The inapproximability of non NP-hard optimization problems. Theoretical Computer Science 289, 553–571 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen, J., Kanj, I., Jia, W.: Vertex cover: further observations and further improvements. Journal of Algorithms 41, 280–301 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen, J., Kanj, I., Xia, G.: Labeled Search Trees and Amortized Analysis: Improved Upper Bounds for NP-Hard Problems. Algorithmica 43, 245–273 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Downey, R.G., Fellows, M.R.: Fixed-parameter intractability. In: Proc. of the 7th Annual Conference on Structure in Complexity Theory, pp. 36–49 (1992)

    Google Scholar 

  15. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness I: Basic results. SIAM Journal on Computing 24, 873–921 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Downey, R.G., Fellows, M.R.: Parameterized Complexity. In: Monographs in Computer Science. Springer, New York (1999)

    Google Scholar 

  17. Downey, R.G., Estivill-Castro, V., Fellows, M.R., Prieto, E., Rosamond, F.A.: Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems. Electr. Notes Theor. Comput. Sci. 78 (2003)

    Google Scholar 

  18. Feige, U., Kilian, J.: On limited versus polynomial nondeterminism. CJTCS: Chicago Journal of Theoretical Computer Science (1997)

    Google Scholar 

  19. Fellows, M.R.: Personal communications (1999) (2005)

    Google Scholar 

  20. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W.H. Freeman and Company, San Francisco (1979)

    MATH  Google Scholar 

  21. Hochbaum, D.: Approximation Algorithms for NP-hard Problems. PWS Publishing (1997)

    Google Scholar 

  22. Kann, V.: On the Approximability of NP-complete Optimization Problems, PhD thesis, Royal Institute of Technology, Sweden (1992)

    Google Scholar 

  23. Karmarkar, N., Karp, R.M.: An efficient approximation scheme for the one-dimensional bin packing problem. In: Proc. of FOCS, pp. 312–320 (1982)

    Google Scholar 

  24. Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal Of Computer and System Sciences 43, 425–440 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  25. Schnorr, C.: Optimal algorithms for self-reducible problems. In: Proc. of ICALP, pp. 322–337 (1976)

    Google Scholar 

  26. Simchi-Levi, D.: New worst-case results for the bin-packing problem. Naval Research Logistics 41, 579–585 (1994)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, The University of Georgia, Athens, Georgia, 30605, USA

    Liming Cai

  2. Department of Computer Science, Arkansas State University, Arkansas, 72467, USA

    Xiuzhen Huang

Authors
  1. Liming Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiuzhen Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University, The Netherlands

    Hans L. Bodlaender

  2. University of Tennessee, 37996-3450, Knoxville, Tennessee, U.S.A.

    Michael A. Langston

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Cai, L., Huang, X. (2006). Fixed-Parameter Approximation: Conceptual Framework and Approximability Results. In: Bodlaender, H.L., Langston, M.A. (eds) Parameterized and Exact Computation. IWPEC 2006. Lecture Notes in Computer Science, vol 4169. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11847250_9

Download citation

  • DOI: https://doi.org/10.1007/11847250_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-39098-5

  • Online ISBN: 978-3-540-39101-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation