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$\Sigma^p_2$-和$\Sigma^p_3$-困难性的证明通常繁琐且困难，目前已知的这两个类的完全问题相对较少，这在最小最大遗憾鲁棒优化、网络阻断、最致命顶点问题、阻碍者问题以及两阶段可调整鲁棒优化领域尤为突出。尽管这些领域已被深入研究二十余年，且人们自然预期其中多数（甚至大多数）问题属于上述类的完全问题，但文献中几乎不存在相应的完全性结果。我们通过引入超过70个全新的$\Sigma^p_2$-完全和$\Sigma^p_3$-完全问题弥补了这一知识空白。我们通过证明一个新的元定理实现了这一成果，该定理同时证明了海量问题类的$\Sigma^p_2$-完全性和$\Sigma^p_3$-完全性。以往上述领域中关于$\Sigma^p_2$-和$\Sigma^p_3$-完全性的绝大多数研究成果均为此元定理的特例。我们精确的结果如下：我们列举了大量适用于该元定理的问题清单（包括团问题、顶点覆盖问题、背包问题、旅行商问题、设施选址问题等）。对清单中的每个问题，我们证明：阻断/最小费用阻碍/最致命节点问题（带元素费用）是$\Sigma^p_2$-完全的；区间不确定下的最小最大遗憾问题是$\Sigma^p_2$-完全的；离散预算不确定下的两阶段可调整鲁棒优化问题是$\Sigma^p_3$-完全的。总之，我们的工作揭示了一个重要洞见：大量NP完全问题具有这样的性质——其最小最大版本"自动"成为$\Sigma^p_2$-完全的。