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.

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