Recoverable robust optimization is a popular multi-stage approach, in which it is possible to adjust a first-stage solution after the uncertain cost scenario is revealed. We consider recoverable robust optimization in combination with discrete budgeted uncertainty. In this setting, it seems plausible that many problems become $\Sigma^p_3$-complete and therefore it is impossible to find compact IP formulations of them (unless the unlikely conjecture NP $= \Sigma^p_3$ holds). Even though this seems plausible, few concrete results of this kind are known. In this paper, we fill that gap of knowledge. We consider recoverable robust optimization for the nominal problems of Sat, 3Sat, vertex cover, dominating set, set cover, hitting set, feedback vertex set, feedback arc set, uncapacitated facility location, $p$-center, $p$-median, independent set, clique, subset sum, knapsack, partition, scheduling, Hamiltonian path/cycle (directed/undirected), TSP, $k$-disjoint path ($k \geq 2$), and Steiner tree. We show that for each of these problems, and for each of three widely used distance measures, the recoverable robust problem becomes $\Sigma^p_3$-complete. Concretely, we show that all these problems share a certain abstract property and prove that this property implies that their robust recoverable counterpart is $\Sigma^p_3$-complete. This reveals the insight that all the above problems are $\Sigma^p_3$-complete 'for the same reason'. Our result extends a recent framework by Gr\"une and Wulf.

翻译：可恢复鲁棒优化是一种流行的多阶段方法，其允许在不确定的成本场景揭示后调整第一阶段的解。我们考虑可恢复鲁棒优化与离散预算不确定性的结合。在此背景下，许多问题似乎很可能成为$\Sigma^p_3$-完全的，因此不可能找到它们的紧凑整数规划公式（除非不太可能的猜想NP $= \Sigma^p_3$成立）。尽管这看似合理，但此类具体结果目前鲜有报道。本文旨在填补这一知识空白。我们针对Sat、3Sat、顶点覆盖、支配集、集合覆盖、命中集、反馈顶点集、反馈弧集、无容量设施选址、$p$-中心、$p$-中值、独立集、团、子集和、背包、划分、调度、哈密顿路径/环（有向/无向）、TSP、$k$-不相交路径（$k \geq 2$）以及斯坦纳树等名义问题，研究其可恢复鲁棒优化形式。我们证明，对于这些问题中的每一个，以及三种广泛使用的距离度量，其可恢复鲁棒问题均变为$\Sigma^p_3$-完全的。具体而言，我们证明所有这些问题共享某种抽象性质，并证明该性质意味着其鲁棒可恢复对应问题是$\Sigma^p_3$-完全的。这揭示了以下洞见：所有上述问题“出于相同原因”都是$\Sigma^p_3$-完全的。我们的结果扩展了Gr\"une和Wulf近期提出的理论框架。