On counting and approximation
- 119 Downloads
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.
Unable to display preview. Download preview PDF.
- E.W.Allender, Invertible functions, Ph.D. Thesis, Georgia Institute of Technology, 1985.Google Scholar
- L. Babai, Trading group theory for randomness, Proc. 17th STOC, 1985, 421–429.Google Scholar
- R.B. Boppana, J. Hastad and S. Zachos, Does co-NP have short interactive proofs?, Information Processing Letters 25, 1987, 127–132.Google Scholar
- A.V. Goldberg and M.Sipser, Compresion and ranking, Proc. 17th STOC, 1985, 440–448.Google Scholar
- S. Goldwasser and M. Sipser, Private coins versus public coins in interactive proof systems, Proc. 18th STOC, 1986, 59–68.Google Scholar
- S. Grollmann and A.L. Selman, Complexity measures for publik-key crypto-systems, Proc. 25th FOCS 1984, 495–503.Google Scholar
- L. Hemachandra, On ranking, Proc. 2nd Structure in Complexity Theory Conf., 1987, 103–117.Google Scholar
- M.W. Krentel, The complexity of optimization problems, Proc. 18th STOC 1986, 69–76.Google Scholar
- M.R. Jerrum, L.G. Valiant and V.V. Vazirani, Random generation of combinatorial structures from a uniform distribution, Theor. Comput. Sci. 43 (1986), 169–188Google Scholar
- U. Schöning, Graph isomorphism is in the low hierarchy, Proc. 4th STACS 1987, Lecture Notes in Comput. Science, 114–124.Google Scholar
- M. Sipser, A complexity theoretic approach to randomness, Proc. 15th STOC, 1983, 330–335.Google Scholar
- L. Stockmeyer, The polynomial time hierarchy, Theoret. Comput. Science 3 (1977), 1–22.Google Scholar
- L. Stockmeyer, On approximation algorithms for #P, SIAM Journ. on Comput. 14 (1985), 849–861.Google Scholar
- L.G. Valiant, Relative complexity of checking and evaluating, Inform. Proc. Letters 5 (1976), 20–23.Google Scholar
- L.G. Valiant, The complexity of computing the permanent, Theor. Comput. Sci. 8 (1979), 189–201.Google Scholar
- L.G. Valiant, The complexity of enumeration and reliability problems, SIAM Journ. Comput. 8 (1979), 410–421.Google Scholar
- K.W. Wagner, Some observations on the connection between counting and recursion, Theor. Comput. Sci. 47 (1986), 131–147.Google Scholar
- S. Zachos, Probabilistic quantifiers, adversaries, and complexity classes: an overview, Proc. Structure in Complexity Theory Conf. 1986, Lecture Notes in Comput Science, Springer, 383–400.Google Scholar
- S. Zachos and M. Fürer, Probabilistic quantifiers vs. distrustful adversaries, manuscript, 1985.Google Scholar