We consider a two-person network inspection game, in which a defender positions a limited number of detectors to detect multiple attacks caused by an attacker. We assume that detection is imperfect, and each detector location is associated with a probability of detecting attacks within its set of monitored network components. The objective of the defender (resp. attacker) is to minimize (resp. maximize) the expected number of undetected attacks. To compute Nash Equilibria (NE) for this large-scale zero-sum game, we formulate a linear program with a small number of constraints, which we solve via column generation. We provide an exact mixed-integer program for the pricing problem, which entails computing a defender's pure best response, and leverage its supermodular structure to derive two efficient approaches to obtain approximate NE with theoretical guarantees: A column generation and a multiplicative weights update (MWU) algorithm with approximate best responses. To address the computational challenges posed by combinatorial attacker strategies, each iteration of our MWU algorithm requires computing a projection under the unnormalized relative entropy. We provide a closed-form solution and a linear-time algorithm for the projection problem. Our computational results in real-world gas distribution networks illustrate the performance and scalability of our solution approaches.

翻译：我们考虑一个双人网络检查博弈，其中防御者部署有限数量的检测器来检测攻击者发起的多次攻击。假设检测是不完美的，每个检测器位置与其监控网络组件集合内检测攻击的概率相关联。防御者（相应地，攻击者）的目标是最小化（相应地，最大化）未被检测到的攻击的期望数量。为计算这一大规模零和博弈的纳什均衡，我们制定了一个约束数量较少的线性规划，并通过列生成方法求解。我们为定价问题提供了一个精确的混合整数规划，该问题涉及计算防御者的纯最佳响应，并利用其超模结构推导出两种具有理论保证的近似纳什均衡高效方法：一种采用近似最佳响应的列生成算法，另一种为乘法权重更新算法。为应对攻击者组合策略带来的计算挑战，我们乘法权重更新算法的每次迭代需在非归一化相对熵下计算投影。我们给出了该投影问题的闭式解与线性时间算法。在真实天然气分配网络上的计算结果表明了我们求解方法的性能与可扩展性。