Computing time-dependent policies for patrolling games with mobile targets

We study how a mobile defender should patrol an area to protect multiple valuable targets from being attacked by an attacker. In contrast to existing approaches, which assume stationary targets, we allow the targets to move through the area according to an a priori known, deterministic movement schedules. We represent the patrol area by a graph of arbitrary topology and do not put any restrictions on the movement schedules. We assume the attacker can observe the defender and has full knowledge of the strategy the defender employs. We construct a game-theoretic formulation and seek defender's optimal randomized strategy in a Stackelberg equilibrium of the game. We formulate the computation of the strategy as a mathematical program whose solution corresponds to an optimal time-dependent Markov policy for the defender. We also consider a simplified formulation allowing only stationary defender's policies which are generally less effective but are computationally significantly cheaper to obtain. We provide experimental evaluation examining this trade-off on a set of test problems covering various topologies of the patrol area and various movement schedules of the targets.