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Abstract

We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class AM. This gives a “PCP characterization” of AM analogous to the PCP Theorem for NP. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the result for AM might be of particular significance for attempts to derandomize this class.

To test this notion, we pose some hypotheses related to our stochastic CSPs that (in light of our result) would imply collapse results for AM. Unfortunately, the hypotheses may be over-strong, and we present evidence against them. In the process we show that, if some language in NP is hard-on-average against circuits of size 2Ω(n), then there exist “inapproximable-on-average” optimization problems of a particularly elegant form.

All our proofs use a powerful form of PCPs known as Probabilistically Checkable Proofs of Proximity, and demonstrate their versatility. We also use known results on randomness-efficient soundness- and hardness-amplification. In particular, we make essential use of the Impagliazzo-Wigderson generator; our analysis relies on a recent Chernoff-type theorem for expander walks.

Keywords

Arthur-Merlin games PCPs average-case complexity 

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References

  1. 1.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. J. ACM 45(3), 501–555 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bellare, M., Goldreich, O., Goldwasser, S.: Randomness in interactive proofs. Computational Complexity 3, 319–354 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.P.: Robust PCPs of proximity, shorter PCPs, and applications to coding. SIAM J. Comput. 36(4), 889–974 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Condon, A., Feigenbaum, J., Lund, C., Shor, P.S.: Probabilistically checkable debate systems and nonapproximability of PSPACE-hard functions. Chicago J. Theor. Comput. Sci (1995)Google Scholar
  5. 5.
    Condon, A., Feigenbaum, J., Lund, C., Shor, P.S.: Random debaters and the hardness of approximating stochastic functions. SIAM J. Comput. 26(2), 369–400 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dinur, I.: The PCP theorem by gap amplification. J. ACM 54(3) (2007)Google Scholar
  7. 7.
    Dinur, I., Reingold, O.: Assignment testers: Towards a combinatorial proof of the PCP theorem. SIAM J. Comput. 36(4), 975–1024 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Drucker, A.: PCPs for Arthur-Merlin Games and Communication Protocols. Master’s thesis, Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science (2010)Google Scholar
  9. 9.
    Haviv, A., Regev, O., Ta-Shma, A.: On the hardness of satisfiability with bounded occurrences in the polynomial-time hierarchy. Theory of Computing 3(1), 45–60 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Healy, A.: Randomness-efficient sampling within NC1. Computational Complexity 17(1), 3–37 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Healy, A., Vadhan, S.P., Viola, E.: Using nondeterminism to amplify hardness. SIAM J. Comput. 35(4), 903–931 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Impagliazzo, R.: Hard-core distributions for somewhat hard problems. In: Proc. 36th IEEE FOCS, pp. 538–545 (1995)Google Scholar
  13. 13.
    Impagliazzo, R., Jaiswal, R., Kabanets, V., Wigderson, A.: Uniform direct product theorems: Simplified, optimized, and derandomized. SIAM J. Comput. 39(4), 1637–1665 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proc. 29th ACM STOC, pp. 220–229 (1997)Google Scholar
  15. 15.
    Ko, K.-I., Lin, C.-C.: Non-approximability in the polynomial-time hierarchy. TR 94-2, Dept. of Computer Science, SUNY at Stony Brook (1994)Google Scholar
  16. 16.
    Mossel, E., Umans, C.: On the complexity of approximating the VC dimension. J. Comput. Syst. Sci. 65(4), 660–671 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    O’Donnell, R.: Hardness amplification within NP. In: Proc. 34th ACM STOC, pp. 751–760 (2002)Google Scholar
  18. 18.
    Shaltiel, R., Umans, C.: Low-end uniform hardness vs. randomness tradeoffs for AM. SIAM J. Comput. 39(3), 1006–1037 (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Wigderson, A., Xiao, D.: A randomness-efficient sampler for matrix-valued functions and applications. In: Proc. 46th IEEE FOCS, pp. 397–406 (2005)Google Scholar
  20. 20.
    Wigderson, A., Xiao, D.: Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators, and applications. Theory of Computing 4(1), 53–76 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Andrew Drucker
    • 1
  1. 1.Computer Science and Artificial Intelligence LaboratoryMassachusetts Institute of TechnologyCambridgeUSA

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