A standard type of uncertainty set in robust optimization is budgeted uncertainty, where an interval of possible values for each parameter is given and the total deviation from their lower bounds is bounded. In the two-stage setting, discrete and continuous budgeted uncertainty have to be distinguished. The complexity of such problems is largely unexplored, in particular if the underlying nominal optimization problem is simple, such as for selection problems. In this paper, we give a comprehensive answer to long-standing open complexity questions for three types of selection problems and three types of budgeted uncertainty sets. In particular, we demonstrate that the two-stage selection problem with continuous budgeted uncertainty is NP-hard, while the corresponding two-stage representative selection problem is solvable in polynomial time. Our hardness result implies that also the two-stage assignment problem with continuous budgeted uncertainty is NP-hard.

翻译：鲁棒优化中一类标准的不确定性集合是预算不确定性，其中每个参数给定一个可能取值区间，且其相对于下界的总偏差有界。在两阶段设定下，必须区分离散与连续预算不确定性。此类问题的复杂性在很大程度上尚未被探索，特别是当基础名义优化问题较为简单时（例如选择类问题）。本文针对三类选择问题与三类预算不确定性集合，对长期悬而未决的复杂性疑问给出了全面解答。特别地，我们证明了具有连续预算不确定性的两阶段选择问题是NP难的，而对应的两阶段代表选择问题可在多项式时间内求解。我们的硬度结果同时表明，具有连续预算不确定性的两阶段分配问题也是NP难的。