Multi-agent patrolling is a key problem in a variety of domains such as intrusion detection, area surveillance, and policing which involves repeated visits by a group of agents to specified points in an environment. While the problem is well-studied, most works do not provide performance guarantees and either do not consider agent attrition or impose significant communication requirements to enable adaptation. In this work, we present the Adaptive Heuristic-based Patrolling Algorithm, which is capable of adaptation to agent loss using minimal communication by taking advantage of Voronoi partitioning, and which meets guaranteed performance bounds. Additionally, we provide new centralized and distributed mathematical programming formulations of the patrolling problem, analyze the properties of Voronoi partitioning, and finally, show the value of our adaptive heuristic algorithm by comparison with various benchmark algorithms using physical robots and simulation based on the Robot Operating System (ROS) 2.