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Abstract: We introduce a 2-round stochastic constraint-satisfaction problem, and showthat its approximation version is complete for the promise version of thecomplexity class AM. This gives a `PCP characterization- of AM analogous to thePCP Theorem for NP. Similar characterizations have been given for higher levelsof the Polynomial Hierarchy, and for PSPACE; however, we suggest that theresult for AM might be of particular significance for attempts to derandomizethis class.To test this notion, we pose some `Randomized Optimization Hypotheses-related to our stochastic CSPs that in light of our result would implycollapse results for AM. Unfortunately, the hypotheses appear over-strong, andwe present evidence against them. In the process we show that, if some languagein NP is hard-on-average against circuits of size 2^{Omegan}, then thereexist hard-on-average optimization problems of a particularly elegant form.All our proofs use a powerful form of PCPs known as ProbabilisticallyCheckable Proofs of Proximity, and demonstrate their versatility. We also useknown results on randomness-efficient soundness- and hardness-amplification. Inparticular, we make essential use of the Impagliazzo-Wigderson generator; ouranalysis relies on a recent Chernoff-type theorem for expander walks.



Author: Andrew Drucker

Source: https://arxiv.org/







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