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，其中代理人在一条直线上具有私人位置，并对两个设施具有公开的审批偏好，其成本是距其批准设施的最大距离。目标是决定设施位置以最小化总成本及最大成本，同时激励代理人保持诚实。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实现了$11$倍和$5$倍近似，从而改进了先前已知的最佳界限$2n+1$和$9$。