Abstract
It is shown that the graph isomorphism problem is located in the low hierarchy in NP. This implies that this problem is not NP-complete (not even under weaker forms of polynomial-time reducibilities, such as γ-reducibility) unless the polynomial-time hierarchy collapses.
Preview
Unable to display preview. Download preview PDF.
References
L. Adleman and K. Manders, Reducibility, randomness, and intractability, Proc. 9th STOC 1977, 151–163.
D. Angluin, On counting problems and the polynomial-time hierarchy, Theor. Comput. Sci. 12 (1980), 161–173.
L. Babai, Trading group theory for randomness, Proc. 17th STOC 1985, 421–429.
L. Babai, Arthur-Merlin games: a randomized proof system and a short hierarchy of complexity classes, manuscript, 1986.
C.H. Bennett and J. Gill, Relative to a random oracle A, pA ≠ NPA ≠ co-NPA with probability 1, SIAM J. Comput. 10 (1981), 96–113.
R.B. Boppana and J. Hastad, Does co-NP have short interactive proofs?, manuscript, 1986.
J.L. Carter and M.N. Wegman, Universal classes of hash functions, J. Comput. Syst. Sci. 18 (1979), 143–154.
M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979.
J. Gill, Computational complexity of probabilistic Turing machines, SIAM J. Comput. 6 (1977), 675–695.
O. Goldreich, S. Micali and A. Widgerson, Proofs that release minimum knowledge, MFCS 84, Lecture Notes in Computer Science 233, Springer, 639–650.
S. Goldwasser and M. Sipser, Private coins versus public coins in interactive proof systems, 18th STOC 1986, 59–68.
C.M. Hoffmann, Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Computer Science 136, Springer, 1982.
K. Ko and U. Schöning, On circuit-size complexity and the low hierarchy in NP, SIAM J. Comput. 14 (1985), 41–51.
C. Lautemann, BPP and the polynomial hierarchy, Inform. Proc. Letters 17 (1983), 215–217.
M. Lerman, Degrees of Unsolvability, Omega-Series, Springer, 1983.
T.J. Long, Strong nondeterministic polynomial-time reducibilities, Theor. Comput. Sci. 21 (1982), 1–25.
R. Mathon, A note on the graph isomorphism counting problem, Inform. Proc. Letters 8 (1979), 131–132.
U. Schöning, A low and a high hierarchy within NP, Journ. Comput. Syst. Sci. 27 (1983), 14–28.
U. Schöning, Complexity and Structure, Lecture Notes in Computer Science 211, Springer, 1986.
M. Sipser, A complexity theoretic approach to randomness, Proc. 15th STOC 1983, 330–335.
L.J. Stockmeyer, The polynomial-time hierarchy, Theor. Comput. Sci. 3 (1977), 1–22.
L.J. Stockmeyer, On approximation algorithms for #P, SIAM J. Comput. 14 (1985), 849–861.
S. Zachos, Probabilistic quantifiers, adversaries, and complexity classes: an overview, Proc. Structure in Complexity Theory Conf., Lecture Notes in Computer Science 223, Springer, 1986, 383–400.
S. Zachos and M. Fürer, Probabilistic quantifiers vs. distrustful adversaries, manuscript, 1985.
S. Zachos and H. Heller, A decisive characterization of BPP, Information and Control 69 (1986), 125–135.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1987 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schöning, U. (1987). Graph isomorphism is in the low hierarchy. In: Brandenburg, F.J., Vidal-Naquet, G., Wirsing, M. (eds) STACS 87. STACS 1987. Lecture Notes in Computer Science, vol 247. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0039599
Download citation
DOI: https://doi.org/10.1007/BFb0039599
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-17219-2
Online ISBN: 978-3-540-47419-7
eBook Packages: Springer Book Archive