We consider a bilevel optimization problem in which the ground set is partitioned between two decision makers, a leader and a follower, whose optimization problems are interleaved. We study the Bilevel Independent Set problem, and its special case, the Bilevel Interval Selection problem, on different variants emerging from a combination of the type of leader's objective function, the type of follower's objective function, and the setting in which the follower reacts, i.e., either optimistically or pessimistically. Here we consider sum and bottleneck type objective functions. We investigate the computational complexity of all these variants for the Bilevel Independent Set problem, and sort them into their respective level of the polynomial hierarchy. Our results range from $\mathsf{P}$, $\mathsf{NP}$-completeness to $Σ_2^\mathsf{p}$-completeness. For the Bilevel Interval Selection problem, we give a dynamic programming algorithm running in time $\mathcal{O}(n^4\log n)$ for the variants in which the leader and the follower have objective functions of the sum type.

翻译：本文研究一类双层优化问题，其中基础集合由两个决策者（领导者和跟随者）划分，两者的优化问题相互交织。我们研究了双层独立集问题及其特例——双层区间选择问题，这些研究基于领导者目标函数类型、跟随者目标函数类型以及跟随者反应方式（即乐观或悲观）组合产生的不同变体。这里我们考虑和型与瓶颈型目标函数。我们探究了双层独立集问题所有变体的计算复杂性，并将其归入多项式层次结构的相应级别。我们的结果包括从$\mathsf{P}$、$\mathsf{NP}$完全到$Σ_2^\mathsf{p}$完全的复杂度范围。对于双层区间选择问题，我们提出了一种动态规划算法，时间复杂度为$\mathcal{O}(n^4\log n)$，适用于领导者和跟随者均采用和型目标函数的变体。