Abstract
We study the class AvgBPP that consists of distributional problems which can be solved in average polynomial time (in terms of Levin’s average-case complexity) by randomized algorithms with bounded error. We prove that there exists a distributional problem that is complete for AvgBPP under polynomial-time samplable distributions. Since we use deterministic reductions, the existence of a deterministic algorithm with average polynomial running time for our problem would imply AvgP = AvgBPP. Note that, while it is easy to construct a promise problem that is complete for \(\bf promise\mbox{-}BPP\) [Mil01], it is unknown whether BPP contains complete languages. We also prove a time hierarchy theorem for AvgBPP (there are no known time hierarchy theorems for BPP). We compare average-case classes with their classical (worst-case) counterparts and show that the inclusions are proper.
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References
Barak, B.: A probabilistic-time hierarchy theorem for slightly non-uniform algorithms. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 194–208. Springer, Heidelberg (2002)
Ben-David, S., Chor, B., Goldreich, O., Luby, M.: On the theory of average case complexity. J. Comput. Syst. Sci. 44(2), 193–219 (1992)
Bogdanov, A., Trevisan, L.: Average-case complexity. Foundation and Trends in Theoretical Computer Science 2(1), 1–106 (2006)
Fortnow, L., Santhanam, R.: Hierarchy theorems for probabilistic polynomial time. In: FOCS, pp. 316–324 (2004)
Grigoriev, D., Hirsch, E.A., Pervyshev, K.: A complete public-key cryptosystem. Technical Report 06-046, Electronic Colloquium on Computational Complexity (2006)
Gurevich, Y.: Average case complexity. In: ICALP, pp. 615–628 (1991)
Hartmanis, J., Hemachandra, L.A.: Complexity classes without machines: On complete languages for up. In: Kott, L. (ed.) ICALP 1986. LNCS, vol. 226, pp. 123–135. Springer, Heidelberg (1986)
Harnik, D., Kilian, J., Naor, M., Reingold, O., Rosen, A.: On robust combiners for oblivious transfer and other primitives. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 96–113. Springer, Heidelberg (2005)
Impagliazzo, R.: A personal view of average-case complexity. In: SCT 1995: Proceedings of the 10th Annual Structure in Complexity Theory Conference (SCT 1995), Washington, DC, USA, p. 134. IEEE Computer Society, Los Alamitos (1995)
Itsykson, D.M.: Structural complexity of AvgBPP. Technical Report 08-073, Electronic Colloquium on Computational Complexity (2008)
Karpinski, M., Verbeek, R.: Randomness, provability, and the separation of Monte Carlo time and space, pp. 189–207. Springer, London (1987)
Levin, L.: Average case complete problems. SIAM Journal on Computing 15(1), 285–286 (1986)
Miltersen, P.B.: Handbook on Randomization, ch. 19. Derandomizing Complexity Classes, vol. II, ch. 19. Kluwer Academic Publishers, Dordrecht (2001)
Pervyshev, K.: On heuristic time hierarchies. In: IEEE Conference on Computational Complexity, pp. 347–358 (2007)
van Melkebeek, D., Pervyshev, K.: A generic time hierarchy with one bit of advice. Computational Complexity 16(2), 139–179 (2007)
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Itsykson, D. (2009). Structural Complexity of AvgBPP. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds) Computer Science - Theory and Applications. CSR 2009. Lecture Notes in Computer Science, vol 5675. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03351-3_16
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DOI: https://doi.org/10.1007/978-3-642-03351-3_16
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