A combinatorial MA-complete problem
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Despite the interest in the complexity class MA, the randomized analog of NP, there is just a couple of known natural (promise-)MA-complete problems, the first due to Bravyi and Terhal (SIAM Journal of Computing 2009) and the second due to Bravyi (Quantum Information and Computation 2015). Surprisingly, both problems are stated using terminology from quantum computation. This fact makes it hard for classical complexity theorists to study these problems, and prevents possible progress, e.g., on the important question of derandomizing MA. In this note we define a natural combinatorial problem called SetCSP and prove its MA-completeness. The problem generalizes the constraint satisfaction problem (CSP) into constraints on sets of strings. This note is, in fact, a combination of previously known works: the brilliant work of Bravyi and Terhal, together with an observation made in our previous work (Aharonov and Grilo, FOCS 2019) that a restricted case of the Bravyi and Terhal's MA complete problem (namely, the uniform case) is already complete, and moreover, that this restricted case can be stated using a classical, combinatorial description. Here we flesh out this observation. This note, along with a translation of the main result of Aharonov and Grilo to the SetCSP language, implies that finding a gap-amplification procedure for SetCSP problems (namely a generalization to SetCSPs of the gap-amplification used in Dinur's PCP proof) would imply MA=NP. This would provide a resolution of the major problem of derandomizing MA; in fact, the problem of finding gap-amplification for SetCSP is in fact equivalent to the MA=NP problem.
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