Many natural optimization problems derived from $\sf NP$ admit bilevel and multilevel extensions in which decisions are made sequentially by multiple players with conflicting objectives, as in interdiction, adversarial selection, and adjustable robust optimization. Such problems are naturally modeled by alternating quantifiers and, therefore, lie beyond $\sf NP$, typically in the polynomial hierarchy or $\sf PSPACE$. Despite extensive study of these problem classes, relatively few natural completeness results are known at these higher levels. We introduce a general framework for proving completeness in the polynomial hierarchy and $\sf PSPACE$ for problems derived from $\sf NP$. Our approach is based on a refinement of $\sf NP$, which we call $\sf NP$ with solutions ($\sf NP$-$\sf S$), in which solutions are explicit combinatorial objects, together with a restricted class of reductions -- solution-embedding reductions -- that preserve solution structure. We define $\sf NP$-$\sf S$-completeness and show that a large collection of classical $\sf NP$-complete problems, including Clique, Vertex Cover, Knapsack, and Traveling Salesman, are $\sf NP$-$\sf S$-complete. Using this framework, we establish general meta-theorems showing that if a problem is $\sf NP$-$\sf S$-complete, then its natural two-level extensions are $Σ_2^p$-complete, its three-level extensions are $Σ_3^p$-complete, and its $k$-level extensions are $Σ_k^p$-complete. When the number of levels is unbounded, the resulting problems are $\sf PSPACE$-complete. Our results subsume nearly all previously known completeness results for multilevel optimization problems derived from $\sf NP$ and yield many new ones simultaneously, demonstrating that high computational complexity is a generic feature of multilevel extensions of $\sf NP$-complete problems.

翻译：许多源于$\sf NP$的自然优化问题允许双层及多层扩展，其中多个目标冲突的参与者依次做出决策，例如阻断问题、对抗选择以及可调鲁棒优化。此类问题自然地通过交替量词建模，因此位于$\sf NP$之外，通常处于多项式层次结构或$\sf PSPACE$中。尽管对这些问题类别已有广泛研究，但在这些更高层次上已知的相对较少的自然完备性结果。我们引入了一个通用框架，用于证明源于$\sf NP$的问题在多项式层次结构和$\sf PSPACE$中的完备性。我们的方法基于对$\sf NP$的一种精化，称为带解的$\sf NP$（$\sf NP$-$\sf S$），其中解是显式的组合对象，并结合一类受限的归约——解嵌入归约——以保持解的结构。我们定义了$\sf NP$-$\sf S$完备性，并证明包括Clique、Vertex Cover、Knapsack和Traveling Salesman在内的大量经典$\sf NP$完备问题是$\sf NP$-$\sf S$完备的。利用此框架，我们建立了通用元定理，表明若一个问题为$\sf NP$-$\sf S$完备，则其自然的两层扩展是$Σ_2^p$完备的，三层扩展是$Σ_3^p$完备的，$k$层扩展是$Σ_k^p$完备的。当层数无界时，所得问题是$\sf PSPACE$完备的。我们的结果涵盖了几乎所有先前已知的源于$\sf NP$的多层优化问题的完备性结果，并同时产生了许多新的结果，表明高计算复杂度是$\sf NP$完备问题的多层扩展的普遍特征。