Despite the interest in the complexity class MA, the randomized analog of NP, just a few natural MA-complete problems are known. The first problem was found by (Bravyi and Terhal, SIAM Journal of Computing 2009); it was then followed by (Crosson, Bacon and Brown, PRE 2010) and (Bravyi, Quantum Information and Computation 2015). Surprisingly, two of these problems are defined using terminology from quantum computation, while the third is inspired by quantum computation and keeps a physical terminology. This prevents classical complexity theorists from studying these problems, delaying potential progress, e.g., on the NP vs. MA question. Here, we define two new combinatorial problems and prove their MA-completeness. The first problem, ACAC, gets as input a succinctly described graph, with some marked vertices. The problem is to decide whether there is a connected component with only unmarked vertices, or the graph is far from having this property. The second problem, SetCSP, generalizes standard constraint satisfaction problem (CSP) into constraints involving sets of strings. Technically, our proof that SetCSP is MA-complete is based on an observation by (Aharonov and Grilo, FOCS 2019), in which it was noted that a restricted case of Bravyi and Terhal's problem (namely, the uniform case) is already MA-complete; a simple trick allows to state this restricted case using combinatorial language. The fact that the first, more natural, problem of ACAC is MA-hard follows quite naturally from this proof, while the containment of ACAC in MA is based on the theory of random walks. We notice that the main result of Aharonov and Grilo carries over to the SetCSP problem in a straightforward way, implying that finding a gap-amplification procedure for SetCSP (as in Dinur's PCP proof) is equivalent to MA=NP. This provides an alternative new path towards the major problem of derandomizing MA.

中文翻译：

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两个组合 MA 完全问题

尽管对复杂性类 MA（NP 的随机模拟）感兴趣，但只有少数自然 MA 完全问题是已知的。第一个问题是由 (Bravyi and Terhal, SIAM Journal of Computing 2009) 发现的；紧随其后的是 (Crosson, Bacon and Brown, PRE 2010) 和 (Bravyi, Quantum Information and Computation 2015)。令人惊讶的是，其中两个问题是使用来自量子计算的术语来定义的，而第三个问题则受到量子计算的启发并保留了物理术语。这阻止了经典复杂性理论家研究这些问题，延迟了潜在的进展，例如，在 NP 与 MA 问题上。在这里，我们定义了两个新的组合问题并证明它们的 MA 完整性。第一个问题 ACAC 将一个简洁描述的图作为输入，其中包含一些标记的顶点。问题是要判断是否存在只有未标记顶点的连通分量，或者图远没有这个属性。第二个问题 SetCSP 将标准约束满足问题 (CSP) 概括为涉及字符串集的约束。从技术上讲，我们证明 SetCSP 是 MA 完全的基于 (Aharonov and Grilo, FOCS 2019) 的观察，其中注意到 Bravyi 和 Terhal 问题的受限情况（即统一情况）已经是 MA-完全的; 一个简单的技巧允许使用组合语言来说明这个受限的情况。从这个证明可以很自然地得出 ACAC 的第一个更自然的问题是 MA-hard 的事实，而 MA 中 ACAC 的包含基于随机游走理论。我们注意到 Aharonov 和 Grilo 的主要结果以一种直接的方式延续到 SetCSP 问题，这意味着找到 SetCSP 的间隙放大程序（如 Dinur 的 PCP 证明）等价于 MA=NP。这为解决 MA 去随机化的主要问题提供了另一种新途径。