Abstract
We show that for every integer k>1, if Σ k , the k’th level of the polynomial-time hierarchy, is worst-case hard for probabilistic polynomial-time algorithms, then there is a language L ∈Σ k 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.
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Gutfreund, D. (2006). Worst-Case Vs. Algorithmic Average-Case Complexity in the Polynomial-Time Hierarchy. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006 2006. Lecture Notes in Computer Science, vol 4110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11830924_36
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DOI: https://doi.org/10.1007/11830924_36
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