We consider a problem where agents are positioned on a line, have approval preferences over two facilities, and their cost is the maximum distance from their approved facilities. The goal is to decide the facility locations to minimize the total and the max cost, while incentivizing the agents to be truthful. We show that a simple strategyproof mechanism is $7$-approximate for the total cost and $5$-approximate for the max cost, thus improving upon the previous bounds of $2n+1$ and $9$.

翻译：我们考虑一个问题：代理人分布在一条直线上，对两个设施有偏好，其成本为到所认可设施的最大距离。目标是在激励代理人诚实的同时，通过决定设施位置来最小化总成本和最大成本。我们证明，一个简单的策略证明机制对于总成本实现7倍近似比，对于最大成本实现5倍近似比，从而改进了此前$2n+1$和$9$的近似比界限。