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Some Results on Average-Case Hardness Within the Polynomial Hierarchy

  • A. Pavan
  • Rahul Santhanam
  • N. V. Vinodchandran
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4337)

Abstract

We prove several results about the average-case complexity of problems in the Polynomial Hierarchy (PH). We give a connection among average-case, worst-case, and non-uniform complexity of optimization problems. Specifically, we show that if PNP is hard in the worst-case then it is either hard on the average (in the sense of Levin) or it is non-uniformly hard (i.e. it does not have small circuits).

Recently, Gutfreund, Shaltiel and Ta-Shma (IEEE Conference on Computational Complexity, 2005) showed an interesting worst-case to average-case connection for languages in NP, under a notion of average-case hardness defined using uniform adversaries. We show that extending their connection to hardness against quasi-polynomial time would imply that NEXP doesn’t have polynomial-size circuits.

Finally we prove an unconditional average-case hardness result. We show that for each k, there is an explicit language in P\(^{\Sigma_2}\) which is hard on average for circuits of size n k .

Keywords

Pseudorandom Generator Hard Function Small Circuit Circuit Family Explicit Language 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • A. Pavan
    • 1
  • Rahul Santhanam
    • 2
  • N. V. Vinodchandran
    • 3
  1. 1.Department of Computer ScienceIowa State University 
  2. 2.Department of Computer ScienceSimon Fraser University 
  3. 3.Department of Computer Science and EngineeringUniversity of Nebraska-Lincon 

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