A PCP Characterization of AM
We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class AM. This gives a “PCP characterization” of AM analogous to the PCP Theorem for NP. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the result for AM might be of particular significance for attempts to derandomize this class.
To test this notion, we pose some hypotheses related to our stochastic CSPs that (in light of our result) would imply collapse results for AM. Unfortunately, the hypotheses may be over-strong, and we present evidence against them. In the process we show that, if some language in NP is hard-on-average against circuits of size 2Ω(n), then there exist “inapproximable-on-average” optimization problems of a particularly elegant form.
All our proofs use a powerful form of PCPs known as Probabilistically Checkable Proofs of Proximity, and demonstrate their versatility. We also use known results on randomness-efficient soundness- and hardness-amplification. In particular, we make essential use of the Impagliazzo-Wigderson generator; our analysis relies on a recent Chernoff-type theorem for expander walks.
KeywordsArthur-Merlin games PCPs average-case complexity
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