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 the leader) and the lower-level decision-maker (referred to as the follower) are given by partition matroids with a common ground set. In contrast to 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 cardinality constraints and engage in a zero-sum game. While a single-level linear integer programming problem over a partition matroid is known to be polynomially solvable, we prove that the considered bilevel problem is NP-hard, even when the objective function coefficients are all binary. On a positive note, it turns out that, if the number of cardinality 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 2-flip local search algorithm for the leader.

翻译：本研究考虑一类线性拟阵阻断问题，其中上层决策者（称为领导者）与下层决策者（称为追随者）的可行集由具有公共基础集的划分拟阵给出。与经典网络阻断模型中领导者受单一预算约束不同，本研究中领导者和追随者均受若干独立基数约束约束，并参与零和博弈。尽管已知划分拟阵上的单层线性整数规划问题可在多项式时间内求解，但本文证明所考虑的双层问题即使目标函数系数均为二进制时仍具有NP-hard性质。值得注意的是，若领导者或追随者的基数约束数量固定，则所考虑的双层问题存在若干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、动态规划方法以及针对领导者的2-翻转局部搜索算法。