We study the computational complexity of multi-stage robust optimization problems. Such multi-stage problems are formulated with alternating min/max quantifiers and therefore naturally fall into higher stage of the polynomial hierarchy. Despite this, almost no hardness result with respect to the polynomial hierarchy are known for robust multi-stage problems. In this work, we examine the hardness of robust two-stage adjustable and robust recoverable optimization with budgeted uncertainty sets. Our main technical contribution is the introduction of a technique tailored to prove $\Sigma^p_3$-hardness of such problems. We highlight a difference between continuous and discrete budgeted uncertainty: In the discrete case, indeed a wide range of problems becomes complete for the third stage of the polynomial hierarchy. We highlight the TSP, independent set, and vertex cover problems as examples of this behavior. However, in the continuous case this does not happen and all problems remain in the first stage of the hierarchy. Finally, if we allow the uncertainty to not only affect the objective, but also multiple constraints, then this distinction disappears and even in the continuous case we encounter hardness for the third stage of the hierarchy. This shows that even robust problems which are already NP-complete can still exhibit a significant computational difference between column-wise and row-wise uncertainty.

翻译：我们研究多阶段强力优化问题的计算复杂性。 这种多阶段问题是用交替的微分/最大量量化器来拟订的,因此自然会进入多元等级的较高阶段。 尽管如此,在多级体系的第三阶段,人们对多级体系几乎没有任何硬性结果,因为存在强有力的多阶段问题。在这项工作中,我们研究强力的两阶段可调整和强力可回收优化的难度,并用预算的不确定性组合进行分析。我们的主要技术贡献是引入一种技术,专门用来证明这类问题是否难于解决问题。我们强调的是连续和单独预算的不确定性之间的差别:在离散的情况下,实际上一系列广泛的问题在多级体系的第三阶段就已经完全完结了。我们强调TSP、独立成套和脊椎将问题作为这种行为的例子。然而,在持续的情况下,这种情况不会发生,所有问题都留在等级体系的第一阶段。最后,如果我们允许这种不确定性不仅影响目标,而且还有多种制约,那么这种区别就会消失,甚至在连续的情况下,甚至在多级等级体系的第三阶段中,我们所遇到的僵硬的等级已经展示了。