We study a new bilevel optimization problem, termed the Randomized Max-Vertex-Cover Interdiction (RMVCI) problem under matroid constraints, which can be modeled as a zero-sum Stackelberg game on a network between a leader and a follower. The leader randomly selects a subset of vertices to protect, subject to a matroid constraint, while the follower-after inferring the leader's protection probability distribution-chooses a subset of vertices (also matroid-constrained) to attack, aiming to maximize the expected total weight of edges incident to the set of vertices that are both attacked and unprotected. The leader's objective is to determine an optimal randomized interdiction strategy that minimizes the follower's expected payoff. Since the follower's response problem is NP-hard, the resulting bilevel program is computationally challenging. We develop a conceptual approximation framework for tackling general bilevel interdiction problems. For the RMVCI problem under matroid constraints, we first formulate the follower's problem as an integer linear program and show that its linear relaxation admits a tight integrality gap of $\tfrac{4}{3}$. Within the approximation framework, we replace the follower's problem by its LP relaxation, and then study the resulting bilevel program. By shifting from distributions over sets to distributions over vertices and applying our approximation framework, we manage to design a polynomial-time 2-approximation algorithm for this relaxed bilevel problem. Combining these ingredients within our framework yields a polynomial-time $\tfrac{8}{3}$-approximation algorithm for RMVCI under matroid constraints.

翻译：我们研究了一种新的双层优化问题，称为基于拟阵约束的随机最大顶点覆盖阻断（RMVCI）问题，该问题可建模为网络中领导者与跟随者之间的零和Stackelberg博弈。领导者在拟阵约束下随机选择保护顶点子集，而跟随者在推断领导者保护概率分布后，选择攻击顶点子集（同样受拟阵约束），旨在最大化被攻击且未受保护的顶点所关联边的期望总权重。领导者的目标是确定最优随机阻断策略，以最小化跟随者的期望收益。由于跟随者的响应问题属于NP难问题，由此产生的双层规划在计算上极具挑战性。我们针对一般双层阻断问题提出了一种概念性近似框架。对于基于拟阵约束的RMVCI问题，我们首先将跟随者问题建模为整数线性规划，并证明其线性松弛具有$\tfrac{4}{3}$的紧整数性间隙。在近似框架内，我们用跟随者问题的线性规划松弛替代原问题，进而研究所得双层规划。通过将基于集合上的概率分布转化为基于顶点上的概率分布，并应用我们的近似框架，成功为该松弛双层问题设计出多项式时间的2-近似算法。结合这些要素，我们的框架最终为基于拟阵约束的RMVCI问题提供了多项式时间的$\tfrac{8}{3}$-近似算法。