Some Results on Average-Case Hardness Within the Polynomial Hierarchy

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 averagecase 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 P2 which is hard on average for circuits of size nk .

• Type: Conference
• Year: 2006
• Where: FSTTCS
• Authors: Aduri Pavan, Rahul Santhanam, N. V. Vinodchandran
• Keywords: Average-case Complexity, Average-case Hardness, FSTTCS 2006, Software Engineering, Unconditional Average-case Hardness