We consider a problem where agents have private positions on a line, and public 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 design a strategyproof mechanism that is simultaneously $11$- and $5$-approximate for these two objective functions, thus improving the previously best-known bounds of $2n+1$ and $9$.

翻译：我们考虑一个问题：智能体在一条直线上拥有私有位置，并对两个设施拥有公开的偏好，其成本是到所批准设施的最大距离。目标是在激励智能体诚实的同时，决定设施位置以最小化总成本和最大成本。我们设计了一个策略证明机制，该机制对这两个目标函数同时具有$11$-近似和$5$-近似性能，从而改进了先前已知的最佳界$2n+1$和$9$。