[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

The parameterized complexity of a problem is generally considered “settled” once it has been shown to be fixed-parameter tractable or to be complete for a class in a parameterized hierarchy such as the weft hierarchy. Several natural parameterized problems have, however, resisted such a classification. In the present paper we argue that, in some cases, this is due to the fact that the parameterized complexity of these problems can be better understood in terms of their parameterized space or parameterized circuit complexity. This includes well-studied, natural problems like the feedback vertex set problem, the associative generability problem, or the longest common subsequence problem. We show that these problems lie in and may even be complete for different parameterized space classes, leading to new insights into the problems’ complexity. The classes we study are defined in terms of different forms of bounded nondeterminism and simultaneous time–space bounds.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bodlaender, H.L.: On linear time minor tests with depth-first search. J. Algorithms 14(1), 1–23 (1993). doi:10.1006/jagm.1993.1001

    Article  MATH  MathSciNet  Google Scholar 

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

  3. Buss, S., Cook, S., Gupta, A., Ramachandran, V.: An optimal parallel algorithm for formula evaluation. SIAM J. Comput. 21(4), 755–780 (1992). doi:10.1137/0221046

    Article  MATH  MathSciNet  Google Scholar 

  4. Buss, S.R.: The Boolean formula value problem is in alogtime. In: Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC 1987), ACM, pp. 123–131. (1987). doi:10.1145/28395.28409

  5. Cai, L., Chen, J., Downey, R.G., Fellows, M.R.: Advice classes of parameterized tractability. Ann. Pure Appl. Log. 84(1), 119–138 (1997a). doi:10.1016/S0168-0072(95)00020-8

    Article  MATH  MathSciNet  Google Scholar 

  6. Cai, L., Chen, J., Downey, R.G., Fellows, M.R.: On the parameterized complexity of short computation and factorization. Arch. Math. Log. 36(4–5), 321–337 (1997b)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, H., Müller, M.: The fine classification of conjunctive queries and parameterized logarithmic space complexity. In: PODS ’13 Proceedings of the 32nd Symposium on Principles of Database Systems, ACM, pp. 309–320. (2013). doi:10.1145/2463664.2463669

  8. Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55(5), 21:2–21:19 (2008). doi:10.1145/1411509.1411511

    Article  MathSciNet  Google Scholar 

  9. Chen, Y., Flum, J., Grohe, M.: Bounded nondeterminism and alternation in parameterized complexity theory. In: Proceedings of the 18th IEEE Annual Conference on Computational Complexity (CCCC 2003), pp. 13–29. (2003). doi:10.1109/ccc.2003.1214407

  10. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)

    Book  Google Scholar 

  11. Elberfeld, M., Stockhusen, C., Tantau, T.: On the space complexity of parameterized problems. In: Thilikos, D.M., Woeginger, G.J. (eds.) Parameterized and Exact Computation, Lecture Notes in Computer Science, vol. 7535, pp. 206–217. Springer, Berlin. (2012).doi:10.1007/978-3-642-33293-7_20

  12. Fellows, M., Hallett, M., Stege, U.: Analogs & duals of the mast problem for squences & trees. J. Algorithms 49(1), 192–216 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Flum, J., Grohe, M.: Describing parameterized complexity classes. Inf. Comput. 187, 291–319 (2003). doi:10.1016/S0890-5401(03)00161-5

    Article  MATH  MathSciNet  Google Scholar 

  14. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006). doi:10.1007/3-540-29953-X

    Google Scholar 

  15. Guillemot, S.: Parameterized complexity and approximability of the longest compatible sequence problem. Discret. Optim. 8(1), 50–60 (2011). doi:10.1016/j.disopt.2010.08.003

    Article  MATH  MathSciNet  Google Scholar 

  16. Hartmanis, J.: On non-determinancy in simple computing devices. Acta Inform. 1, 336–344 (1972). doi:10.1007/BF00289513

    Article  MATH  MathSciNet  Google Scholar 

  17. Hong, J.W.: On some deterministic space complexity problems. SIAM J. Comput. 11(3), 591–601 (1982). doi:10.1137/0211050

    Article  MATH  MathSciNet  Google Scholar 

  18. Immerman, N.: Descriptive Complexity. Springer, New York (1998)

    Google Scholar 

  19. Jones, N.D., Laaser, W.T.: Complete problems for deterministic polynomial time. Theoret. Comput. Sci. 3(1), 105–117 (1976). doi:10.1016/0304-3975(76)90068-2

    Article  MathSciNet  Google Scholar 

  20. Jones, N.D., Lien, Y.E., Laaser, W.T.: New problems complete for nondeterministic log space. Math. Syst. Theory 10(1), 1–17 (1976). doi:10.1007/BF01683259

    Article  MATH  MathSciNet  Google Scholar 

  21. Kintala, C.M.R., Fischer, P.C.: Refining nondeterminism in relativized polynomial-time bounded computations. SIAM J. Comput. 1(9), 46–53 (1980). doi:10.1137/0209003

    Article  MathSciNet  Google Scholar 

  22. Stockhusen, C., Tantau, T.: Completeness results for parameterized space classes. Tech. Rep. arXiv:1308.2892, ArXiv e-prints (2013a)

  23. Stockhusen, C., Tantau, T.: Completeness results for parameterized space classes. In: Gutin, G., Szeider, S. (eds.) Parameterized and Exact Computation, Lecture Notes in Computer Science, vol. 8246, pp. 335–347. Springer, Berlin (2013b). doi:10.1007/978-3-319-03898-8_28

    Chapter  Google Scholar 

  24. Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, Berlin (1999)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chair of Computer Science 7, RWTH Aachen University, 52056, Aachen, Germany

    Michael Elberfeld

  2. Institute for Theoretical Computer Science, Universität zu Lübeck, 23538, Lübeck, Germany

    Christoph Stockhusen & Till Tantau

Authors
  1. Michael Elberfeld
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Stockhusen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Till Tantau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christoph Stockhusen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elberfeld, M., Stockhusen, C. & Tantau, T. On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness. Algorithmica 71, 661–701 (2015). https://doi.org/10.1007/s00453-014-9944-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-014-9944-y

Keywords

Search

Navigation