Advertisement

Acta Informatica

, Volume 26, Issue 4, pp 363–379 | Cite as

On counting and approximation

  • Johannes Köbler
  • Uwe Schöning
  • Jacobo Toran
Article
  • 88 Downloads

Summary

We introduce a new class of functions, called span functions which count the different output values that occur at the leaves of the computation tree associated with a nondeterministic polynomial time Turing machine transducer. This function class has natural complete problems; it is placed between Valiant's function classes # P and # NP, and contains both Goldberg and Sipser's ranking functions for sets in NP, and Krentel's optimization functions. We show that it is unlikely that the span functions coincide with any of the mentioned function classes. A probabilistic approximation method (using an oracle in NP) is presented to approximate span functions up to any desired degree of accuracy. This approximation method is based on universal hashing and it never underestimates the correct value of the approximated function.

Keywords

Approximation Method Computational Mathematic Computer System System Organization Polynomial Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allender, E.W.: Invertible functions. Ph.D. Thesis, Georgia Institute of Technology 1985Google Scholar
  2. 2.
    Babai, L.: Trading group theory for randomness. Proc. 17th STOC, 1985, pp. 421–429Google Scholar
  3. 3.
    Boppana, R.B., Hastad, J., Zachos, S.: Does co-NP have short interactive proofs? Inf. Process. Lett. 25, 127–132 (1987)Google Scholar
  4. 4.
    Goldberg, A.V., Sipser, M.: Compression and ranking. Proc. 17th STOC, 1985, pp. 440–448Google Scholar
  5. 5.
    Goldwasser, S., Sipser, M.: Private coins versus public coins in interactive proof systems. Proc. 18th STOC, 1986, pp. 59–68Google Scholar
  6. 6.
    Grollmann, S., Selman, A.L.: Complexity measures for public-key crypto-systems. 25th FOCS 1984, pp. 495–503Google Scholar
  7. 7.
    Hemachandra, L.: On ranking. Proc. 2nd Structure in Complexity Theory Conf., 1987, pp. 103–117Google Scholar
  8. 8.
    Jerrum, M.R., Valiant, L.G., Vazirani, V.V.: Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci. 43, 169–188 (1986)Google Scholar
  9. 9.
    Karp, R.M.: Reducibility among combinatorial problems. Proc. Symp. Complexity Computation 1972Google Scholar
  10. 10.
    Krentel, M.W.: The complexity of optimization problems. Proc. 18th STOC 1986, pp. 69–76Google Scholar
  11. 11.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial optimization. New Jersey: Prentice-Hall 1982Google Scholar
  12. 12.
    Schöning, U.: Graph isomorphism is in the low hierarchy. Proc. 4th STACS 1987. (Lect. Notes Comput. Sci., pp. 114–124)Google Scholar
  13. 13.
    Sipser, M.: A complexity theoretic approach to randomness. Proc. 15th STOC, 1983, pp. 330–335Google Scholar
  14. 14.
    Stockmeyer, L.: The polynomial time hierarchy. Theor. Comput. Sci. 3, 1–22 (1977)Google Scholar
  15. 15.
    Stockmeyer, L.: On approximation algorithms for #P. SIAM J. Comput. 14, 849–861 (1985)Google Scholar
  16. 16.
    Valiant, L.G.: Relative complexity of checking and evaluating. Inf. Process. Lett. 5, 20–23 (1976)Google Scholar
  17. 17.
    Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)Google Scholar
  18. 18.
    Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 410–421 (1979)Google Scholar
  19. 19.
    Wagner, K.: Some observations on the connection between counting and recursion. Theor. Comput. Sci. 47, 131–147 (1986)Google Scholar
  20. 20.
    Zachos, S.: Probabilistic quantifiers, adversaries, and complexity classes: an overview. Proc. Struct. Complexity Theory Conf. 1986 (Lect. Notes Comput. Sci., vol. 233, pp. 383–400). Berlin Heidelberg New York: SpringerGoogle Scholar
  21. 21.
    Zachos, S., Fürer, M.: Probabilistic quantifiers vs. distrustful adversaries, (manuscript, 1985)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Johannes Köbler
    • 1
  • Uwe Schöning
    • 2
  • Jacobo Toran
    • 3
  1. 1.Universität StuttgartStuttgart 1Federal Republic of Germany
  2. 2.EWH KoblenzKoblenzFederal Republic of Germany
  3. 3.Facultat d'InformaticaBarcelonaSpain

Personalised recommendations