Strategy Synthesis in Adversarial Patrolling Games

Patrolling is one of the central problems in operational security. Formally, a patrolling problem is specified by a set $U$ of nodes (admissible defender's positions), a set $T \subseteq U$ of vulnerable targets, an admissible defender's moves over $U$, and a function which to every target assigns the time needed to complete an intrusion at it. The goal is to design an optimal strategy for a defender who is moving from node to node and aims at detecting possible intrusions at the targets. The goal of the attacker is to maximize the probability of a successful attack. We assume that the attacker is adversarial, i.e., he knows the strategy of the defender and can observe her moves. We prove that the defender has an optimal strategy for every patrolling problem. Further, we show that for every $\varepsilon$ > 0, there exists a finite-memory $\varepsilon$-optimal strategy for the defender constructible in exponential time, and we observe that such a strategy cannot be computed in polynomial time unless P=NP. Then we focus ourselves to unrestricted defender's moves. Here, a patrolling problem is fully determined by its signature, the number of targets of each attack length. We bound the maximal probability of successfully defended attacks. Then, we introduce a decomposition method which allows to split a given patrolling problem $G$ into smaller subproblems and construct a defender's strategy for $G$ by "composing" the strategies constructed for these subproblems. Finally, for patrolling problems with $T = U$ and a well-formed signature, we give an exact classification of all sufficiently connected environments where the defender can achieve the same value as in the fully connected uniform environment. This result is useful for designing "good" environments where the defender can act optimally.