In the well-known complexity class NP are combinatorial problems, whose optimization counterparts are important for many practical settings. These problems typically consider full knowledge about the input. In practical settings, however, uncertainty in the input data is a usual phenomenon, whereby this is normally not covered in optimization versions of NP problems. One concept to model the uncertainty in the input data, is recoverable robustness. The instance of the recoverable robust version of a combinatorial problem P is split into a base scenario $\sigma_0$ and an uncertainty scenario set $\textsf{S}$. The base scenario and all members of the uncertainty scenario set are instances of the original combinatorial problem P. The task is to calculate a solution $s_0$ for the base scenario $\sigma_0$ and solutions $s$ for all uncertainty scenarios $\sigma \in \textsf{S}$ such that $s_0$ and $s$ are not too far away from each other according to a distance measure, so $s_0$ can be easily adapted to $s$. This paper introduces Hamming Distance Recoverable Robustness, in which solutions $s_0$ and $s$ have to be calculated, such that $s_0$ and $s$ may only differ in at most $\kappa$ elements. We survey the complexity of Hamming distance recoverable robust versions of optimization problems, typically found in NP for different scenario encodings. The complexity is primarily situated in the lower levels of the polynomial hierarchy. The main contribution of the paper is a gadget reduction framework that shows that the recoverable robust versions of problems in a large class of combinatorial problems is $\Sigma^P_{3}$-complete. This class includes problems such as Vertex Cover, Coloring or Subset Sum. Additionally, we expand the results to $\Sigma^P_{2m+1}$-completeness for multi-stage recoverable robust problems with $m \in \mathbb{N}$ stages.

翻译：在著名的复杂度类NP中包含组合问题，其优化对应版本对许多实际场景具有重要意义。这些问题通常假设对输入具有完全知识。然而在实际场景中，输入数据的不确定性是常见现象，而这通常未被NP问题的优化版本所覆盖。可恢复鲁棒性是一种建模输入数据不确定性的概念。组合问题P的可恢复鲁棒版本的实例被划分为基准场景$\sigma_0$和不确定性场景集$\textsf{S}$。基准场景及不确定性场景集的所有成员均为原始组合问题P的实例。任务是计算基准场景$\sigma_0$的解$s_0$以及所有不确定性场景$\sigma \in \textsf{S}$的解$s$，使得$s_0$与$s$根据某种距离度量彼此偏离不超过一定范围，从而$s_0$可被便捷地调整为$s$。本文引入汉明距离可恢复鲁棒性，要求计算解$s_0$与$s$，使得二者最多在$\kappa$个元素上存在差异。我们系统研究了优化问题的汉明距离可恢复鲁棒版本的复杂度（这些问题的原始版本通常属于NP），并针对不同场景编码进行分析。其复杂度主要位于多项式层次结构的低层级中。本文的主要贡献在于提出了一种基于部件归约的框架，证明一大类组合问题的可恢复鲁棒版本属于$\Sigma^P_{3}$-完全类。该类问题包括顶点覆盖、图着色或子集和等经典问题。此外，我们将结果推广至多阶段可恢复鲁棒问题，证明其对于$m \in \mathbb{N}$个阶段具有$\Sigma^P_{2m+1}$-完全性。