Because $\Sigma^p_2$- and $\Sigma^p_3$-hardness proofs are usually tedious and difficult, not so many complete problems for these classes are known. This is especially true in the areas of min-max regret robust optimization, network interdiction, most vital vertex problems, blocker problems, and two-stage adjustable robust optimization problems. Even though these areas are well-researched for over two decades and one would naturally expect many (if not most) of the problems occurring in these areas to be complete for the above classes, almost no completeness results exist in the literature. We address this lack of knowledge by introducing over 70 new $\Sigma^p_2$-complete and $\Sigma^p_3$-complete problems. We achieve this result by proving a new meta-theorem, which shows $\Sigma^p_2$- and $\Sigma^p_3$-completeness simultaneously for a huge class of problems. The majority of all earlier publications on $\Sigma^p_2$- and $\Sigma^p_3$-completeness in said areas are special cases of our meta-theorem. Our precise result is the following: We introduce a large list of problems for which the meta-theorem is applicable (including clique, vertex cover, knapsack, TSP, facility location and many more). For every problem on this list, we show: The interdiction/minimum cost blocker/most vital nodes problem (with element costs) is $\Sigma^p_2$-complete. The min-max-regret problem with interval uncertainty is $\Sigma^p_2$-complete. The two-stage adjustable robust optimization problem with discrete budgeted uncertainty is $\Sigma^p_3$-complete. In summary, our work reveals the interesting insight that a large amount of NP-complete problems have the property that their min-max versions are 'automatically' $\Sigma^p_2$-complete.

翻译：由于$\Sigma^p_2$-与$\Sigma^p_3$-困难性证明通常繁琐且困难，目前已知的这些复杂性类的完备问题数量有限。这一现象在最小最大遗憾鲁棒优化、网络阻断、最关键顶点问题、阻塞器问题以及两阶段可调鲁棒优化等领域尤为显著。尽管这些领域已被深入研究超过二十年，且人们自然会预期其中出现的问题（即使不是大多数）对上述复杂性类具有完备性，但文献中几乎不存在完备性结果。我们通过引入70余个新的$\Sigma^p_2$-完备与$\Sigma^p_3$-完备问题来填补这一知识空白。这一成果的取得依赖于我们证明的一个新元定理，该定理能同时为一大类问题建立$\Sigma^p_2$-与$\Sigma^p_3$-完备性。上述领域早期关于$\Sigma^p_2$-与$\Sigma^p_3$-完备性的文献成果大多是我们元定理的特例。我们的具体成果如下：我们列出了一系列可应用元定理的问题（包括团、顶点覆盖、背包问题、旅行商问题、设施选址等众多问题）。对于列表中的每个问题，我们证明：带元素成本的阻断/最小成本阻塞器/最关键节点问题是$\Sigma^p_2$-完备的；区间不确定下的最小最大遗憾问题是$\Sigma^p_2$-完备的；离散预算不确定下的两阶段可调鲁棒优化问题是$\Sigma^p_3$-完备的。总而言之，我们的工作揭示了一个深刻见解：大量NP完备问题具有这样的性质——它们的最小最大版本会"自动"成为$\Sigma^p_2$-完备问题。