Some results on average-case hardness within the polynomial hierarchy

  • Authors:

  • A. Pavan;Rahul Santhanam;N. V. Vinodchandran

  • Affiliations:

  • Department of Computer Science, Iowa State University;Department of Computer Science, Simon Fraser University;Department of Computer Science and Engineering, University of Nebraska-Lincon

  • Venue:
  • FSTTCS'06 Proceedings of the 26th international conference on Foundations of Software Technology and Theoretical Computer Science
  • Year:
  • 2006

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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 nk.