We study the computational complexity of multi-stage robust optimization problems. Such problems are formulated with alternating min/max quantifiers and therefore naturally fall into a higher stage of the polynomial hierarchy. Despite this, almost no hardness results with respect to the polynomial hierarchy are known. 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; in particular, this applies to the TSP, independent set, and vertex cover problems. However, in the continuous case this does not happen and 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：我们研究多阶段鲁棒优化问题的计算复杂性。这类问题以交替的 min/max 量词表述，因此自然属于多项式层级的更高阶段。尽管如此，目前几乎没有任何关于多项式层级难度的已知结论。本文分析了具有预算不确定性集合的鲁棒两阶段可调优化问题和鲁棒可恢复优化问题的难解性。我们的主要技术贡献是引入一种专门用于证明此类问题为 $\Sigma^p_3$-困难的技巧。我们突出了离散型与连续型预算不确定性之间的区别：在离散情形下，广泛的问题（包括旅行商问题、独立集问题和顶点覆盖问题）确实成为多项式层级第三阶段的完全问题。然而，在连续情形下，这种情况不会发生，问题仍保持在多项式层级的第一阶段。最后，若允许不确定性不仅影响目标函数，还影响多个约束条件，则该区别消失，即使在连续情形下我们也会遇到多项式层级第三阶段的难解性。这表明，即使是已经属于 NP-完全的鲁棒问题，在列不确定性与行不确定性之间仍可能存在显著的计算差异。