In full-knowledge multi-robot adversarial patrolling, a group of robots has to detect an adversary who knows the robots' strategy. The adversary can easily take advantage of any deterministic patrolling strategy, which necessitates the employment of a randomised strategy. While the Markov decision process has been the dominant methodology in computing the penetration detection probabilities on polylines, we apply enumerative combinatorics to characterise the penetration detection probabilities for four penetration configurations. It allows us to provide the closed formulae of these probabilities and facilitates characterising optimal random defence strategies. Comparing to iteratively updating the Markov transition matrices, we empirically show that our method reduces the runtime by up to several hours. This allows us extensive simulations on the two dominant robot movement types for patrolling a perimeter showing that a movement with direction is up to 0.4 more likely to detect an adversary. Therefore, our approach greatly benefits the theoretical and empirical analysis of optimal patrolling strategies with extendability to more complicated attacks and other structured environments.

在全知识多机器人对抗巡逻中，一组机器人必须检测一个知道机器人策略的对手。对手可以轻松利用任何确定性巡逻策略，这需要采用随机策略。虽然马尔可夫决策过程一直是计算折线上渗透检测概率的主要方法，但我们应用了枚举组合学来表征四种穿透配置的穿透检测概率。它使我们能够提供这些概率的封闭公式，并有助于表征最优随机防御策略。与迭代更新马尔可夫转移矩阵相比，我们凭经验表明我们的方法将运行时间缩短了几个小时。这使我们能够对两种主要的机器人运动类型进行广泛的模拟，以在周边巡逻，表明有方向的运动检测到对手的可能性高达 0.4。因此，我们的方法极大地有利于对最优巡逻策略的理论和实证分析，并可以扩展到更复杂的攻击和其他结构化环境。