In this study, we consider a class of linear matroid interdiction problems, where the feasible sets for the upper-level decision-maker (referred to as a leader) and the lower-level decision-maker (referred to as a follower) are induced by two distinct partition matroids with a common weighted ground set. Unlike classical network interdiction models where the leader is subject to a single budget constraint, in our setting, both the leader and the follower are subject to several independent capacity constraints and engage in a zero-sum game. While the problem of finding a maximum weight independent set in a partition matroid is known to be polynomially solvable, we prove that the considered bilevel problem is $NP$-hard even when the weights of ground elements are all binary. On a positive note, it is revealed that, if the number of capacity constraints is fixed for either the leader or the follower, then the considered class of bilevel problems admits several polynomial-time solution schemes. Specifically, these schemes are based on a single-level dual reformulation, a dynamic programming-based approach, and a greedy algorithm for the leader.

翻译：在本研究中，我们考虑一类线性拟阵拦截问题，其中上层决策者（称为领导者）和下层决策者（称为跟随者）的可行集由两个具有公共加权基础集的划分拟阵诱导而成。与经典网络拦截模型中领导者受单一预算约束不同，在我们的设定中，领导者和跟随者均受到多个独立的容量约束，并参与一个零和博弈。尽管已知在划分拟阵中寻找最大权独立集问题是多项式时间可解的，我们证明了即使基础元素权重均为二元时，所考虑的双层问题也是$NP$-难的。从积极方面看，研究表明，若领导者或跟随者的容量约束数量固定，则该类双层问题存在多种多项式时间求解方案。具体而言，这些方案基于单层对偶重构、一种基于动态规划的方法以及一种针对领导者的贪心算法。