Worst-case vs. algorithmic average-case complexity in the polynomial-time hierarchy

Authors:
Dan Gutfreund
Affiliations:
Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA
Venue:
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Year:
2006

Citing 13
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Abstract

We show that for every integer k1, if Σk, the k'th level of the polynomial-time hierarchy, is worst-case hard for probabilistic polynomial-time algorithms, then there is a language L ∈Σk such that for every probabilistic polynomial-time algorithm that attempts to decide it, there is a samplable distribution over the instances of L, on which the algorithm errs with probability at least 1/2–1/poly(n) (where the probability is over the choice of instances and the randomness of the algorithm). In other words, on this distribution the algorithm essentially does not perform any better than the algorithm that simply decides according to the outcome of an unbiased coin toss.