Patrol Scheduling Against Adversaries with Varying Attack Durations

We consider a generalization of zero-sum patrolling security game that allows the attacker choosing when, where, and how long to launch an attack, under three different attacker models. The attacker's payoff is the acquired utilities of the attack minus a penalty if the attacker is caught by the defender in patrol. The goal is to reduce the payoff of the attacker. To find the optimal defender/ attacker strategy, the game is converted to a combinatorial minimax problem with a closed-form objective function. Due to the complexity of the utility functions, we show that the minimax problem is not convex for all attacker models, even when the defender strategy is assumed as the time-homogeneous first-order Markov chain (i.e., the patroller's next visit only depends on his current location). However, for the zero penalty case, we prove that the optimal solution is either minimizing the expected hitting time or return time, based on different attacker models. We also observe that increasing the randomness of the patrol schedule helps to reduce the attacker's expected payoff for high penalty cases. Thus, to find solutions for general cases, we formulated a bi-criteria optimization problem and proposed three algorithms that support finding a trade-off between the expected maximum reward and the randomness. Another characteristic is that the third algorithm is able to find the optimal deterministic patrol schedule, although the running time is exponential on the number of patrol spots. Experiments demonstrate the effectiveness and scalability of our solutions. It also shows that our solutions outperform the baselines from state of the art in both artificial and real-world crime datasets.