Recoverable robust optimization is a multi-stage approach, where it is possible to adjust a first-stage solution after the uncertain cost scenario is revealed. We analyze this approach for a class of selection problems. The aim is to choose a fixed number of items from several disjoint sets, such that the worst-case costs after taking a recovery action are as small as possible. The uncertainty is modeled as a discrete budgeted set, where the adversary can increase the costs of a fixed number of items. While special cases of this problem have been studied before, its complexity has remained open. In this work we make several contributions towards closing this gap. We show that the problem is NP-hard and identify a special case that remains solvable in polynomial time. We provide a compact mixed-integer programming formulation and two additional extended formulations. Finally, computational results are provided that compare the efficiency of different exact solution approaches.

中文翻译：

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具有离散预算不确定性的可恢复鲁棒代表选择问题

可恢复的鲁棒优化是一种多阶段方法，可以在显示不确定的成本场景后调整第一阶段的解决方案。我们针对一类选择问题分析了这种方法。目的是从几个不相交的集合中选择固定数量的项目，以便采取恢复行动后的最坏情况成本尽可能小。不确定性被建模为一个离散的预算集，其中对手可以增加固定数量项目的成本。虽然之前已经研究过这个问题的特殊情况，但它的复杂性仍然开放。在这项工作中，我们为缩小这一差距做出了一些贡献。我们证明该问题是 NP-hard 问题，并确定了一个在多项式时间内仍可解决的特殊情况。我们提供了一个紧凑的混合整数规划公式和两个额外的扩展公式。最后，提供了比较不同精确求解方法的效率的计算结果。