We study the complexity of motivating time-inconsistent agents to complete long term projects in a graph-based planning model proposed by Kleinberg and Oren (2014). Given a task graph G with n nodes, our objective is to guide an agent towards a target node t under certain budget constraints. The crux is that the agent may change its strategy over time due to its present-bias. We consider two strategies to guide the agent. First, a single reward is placed at t and arbitrary edges can be removed from G. Secondly, rewards can be placed at arbitrary nodes of G but no edges must be deleted. In both cases we show that it is NP-complete to decide if a given budget is sufficient to keep the agent motivated. For the first setting, we give complementing upper and lower bounds on the approximability of the minimum required budget. In particular, we devise a \((1+\sqrt {n})\)-approximation algorithm and prove NP-hardness for ratios greater than \(\sqrt {n}/3\). We also argue that the second setting does not permit any efficient approximation unless P = NP.

中文翻译：

---

激励时间不一致的代理人：一种计算方法。

我们在由Kleinberg和Oren（2014）提出的基于图的计划模型中研究了激励时间不一致的代理商完成长期项目的复杂性。给定一个具有n个节点的任务图G，我们的目标是在一定的预算约束下将代理引导到目标节点t。问题的关键是，代理可能会由于其当前偏见而随时间改变其策略。我们考虑两种策略来指导代理商。首先，将单个奖励置于t处，并且可以从G移除任意边。其次，可以将奖励放置在G的任意节点上但不得删除任何边缘。在这两种情况下，我们都表明，确定给定的预算是否足以使代理商保持动力是NP完全的。对于第一个设置，我们给出了最低所需预算的近似性的互补上限和下限。特别是，我们设计了一种\（（1+ \ sqrt {n}）\）近似算法，并证明了比率大于\（\ sqrt {n} / 3 \）的NP硬度。我们还认为，除非P = NP，否则第二设置不允许任何有效逼近。