Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Parameterized and Exact Computation

IPEC 2011: Parameterized and Exact Computation pp 169–180Cite as

Safe Approximation and Its Relation to Kernelization

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7112))

Abstract

We introduce a notion of approximation, called safe approximation, for minimization problems that are subset problems. We first study the relation between the standard notion of approximation and safe approximation, and show that the two notions are different unless some unlikely collapses in complexity theory occur. We then study the relation between safe approximation and kernelization. We demonstrate how the notion of safe approximation can be useful in designing kernelization algorithms for certain fixed-parameter tractable problems. On the other hand, we show that there are problems that have constant-ratio safe approximation algorithms but no polynomial kernels, unless the polynomial hierarchy collapses to the third level.

This is a preview of subscription content, 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   54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alber, J., Fellows, M., Niedermeier, R.: Polynomial-time data reduction for dominating set. J. ACM 51(3), 363–384 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the Undirected Feedback Vertex Set problem. SIAM J. Discrete Math. 12(3), 289–297 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the Weighted Vertex Cover problem. Annals of Discrete Mathematics 25, 27–46 (1985)

    MathSciNet  MATH  Google Scholar 

  5. Bar-Yehuda, R., Geiger, D., Naor, J., Roth, R.: Approximation algorithms for the Feedback Vertex Set problem with applications to constraint satisfaction and bayesian inference. SIAM J. Comput. 27(4), 942–959 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bodlaender, H.L., Penninkx, E.: A Linear Kernel for Planar Feedback Vertex Set. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 160–171. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  7. Buss, J., Goldsmith, J.: Nondeterminism within P. SIAM J. Comput. 22, 560–572 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cai, L., Chen, J.: Fixed parameter tractability and approximability of NP-hard optimization problems. J. Comput. Syst. Sci. 54, 465–474 (1997)

    Article  MATH  Google Scholar 

  9. Cai, L., Fellows, M., Juedes, D., Rosamond, F.: The complexity of polynomial-time approximation. Theory Comput. Syst. 41(3), 459–477 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Inf. Process. Lett. 64, 165–171 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, J., Fernau, H., Kanj, I., Xia, G.: Parametric duality and kernelization: Lower bounds and upper bounds on kernel size. SICOMP 37(4), 1077–1106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, J., Huang, X., Kanj, I., Xia, G.: Linear FPT reductions and computational lower bounds. J. Comput. Syst. Sci. 72(8), 1346–1367 (2006)

    Article  MATH  Google Scholar 

  13. Chen, J., Huang, X., Kanj, I., Xia, G.: Polynomial time approximation schemes and parameterized complexity. Discrete Appl. Mathematics 155(2), 180–193 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: STOC, pp. 251–260 (2010)

    Google Scholar 

  16. Downey, R., Fellows, M.: Parameterized Complexity. Springer, New York (1999)

    Book  MATH  Google Scholar 

  17. Johnson, D.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)

    Article  MATH  Google Scholar 

  18. Kanj, I., Pelsmajer, M., Xia, G., Schaefer, M.: On the induced matching problem. In: STACS. LIPIcs, vol. 08001, pp. 397–408 (2008)

    Google Scholar 

  19. Kolaitis, P., Thakur, M.: Approximation properties of NP minimization classes. J. Comput. Syst. Sci. 50, 391–411 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kratsch, S.: Polynomial kernelizations for MIN F + Π1 and MAX NP. In: STACS. LIPIcs, vol. 3, pp. 601–612 (2009)

    Google Scholar 

  21. Marx, D.: Efficient Approximation Schemes for Geometric Problems? In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 448–459. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  22. Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)

    Article  Google Scholar 

  23. Nemhauser, G., Trotter, L.: Vertex packing: structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Savage, C.: Depth-first search and the vertex cover problem. Inf. Process. Lett. 14(5), 233–237 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Thomassé, S.: A 4k 2 kernel for feedback vertex set. ACM Transactions on Algorithms 6(2) (2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität des Saarlandes, Saarbrücken, Germany

    Jiong Guo

  2. DePaul University, Chicago, USA

    Iyad Kanj

  3. Utrecht University, Utrecht, The Netherlands

    Stefan Kratsch

Authors
  1. Jiong Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iyad Kanj
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Kratsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. MTA SZTAKI, Lágymányosi u. 11, 1111, Budapest, Hungary

    Dániel Marx

  2. Lehrgebiet Theoretische Informatik, RWTH Aachen, Ahornstraße 55, 52056, Aachen, Germany

    Peter Rossmanith

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Guo, J., Kanj, I., Kratsch, S. (2012). Safe Approximation and Its Relation to Kernelization. In: Marx, D., Rossmanith, P. (eds) Parameterized and Exact Computation. IPEC 2011. Lecture Notes in Computer Science, vol 7112. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28050-4_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-28050-4_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28049-8

  • Online ISBN: 978-3-642-28050-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation