Consider oriented graph nodes requiring periodic visits by a service agent. The agent moves among the nodes and receives a payoff for each completed service task, depending on the time elapsed since the previous visit to a node. We consider the problem of finding a suitable schedule for the agent to maximize its long-run average payoff per time unit. We show that the problem of constructing an $\varepsilon$-optimal schedule is PSPACE-hard for every fixed $\varepsilon \geq 0$, and that there exists an optimal periodic schedule of exponential length. We propose randomized finite-memory (RFM) schedules as a compact description of the agent's strategies and design an efficient algorithm for constructing RFM schedules. Furthermore, we construct deterministic periodic schedules by sampling from RFM schedules.

翻译：考虑有向图节点需要服务代理进行周期性访问。代理在节点间移动，并根据自上次访问某节点以来经过的时间，为每次完成的服务任务获得收益。我们研究如何为该代理寻找合适的调度方案，以最大化其每单位时间内的长期平均收益。我们证明，对于任意固定的 $\varepsilon \geq 0$，构建 $\varepsilon$-最优调度的问题是PSPACE-难的，并且存在一个指数长度的最优周期调度。我们提出随机有限记忆（RFM）调度作为代理策略的紧凑描述，并设计了一种高效算法来构建RFM调度。此外，我们通过对RFM调度进行采样，构建了确定性周期调度。