AI 中文总结
研究平等主义目标下战略设施选址问题，证明防策略机制近似比渐近下界，给出双代理随机近似机制；考虑输出增强框架，设计不同设置下的确定性或随机近似机制，展现输出增强可替代随机性及成组防策略特性。
AI 中文摘要
我们研究了平等主义目标下的战略设施选址问题，即一种机制利用欧几里得空间中一组代理报告的位置来选择一个设施位置，使到任何代理的最大距离最小化。我们将注意力限制在防策略机制上，确保没有代理能通过误报位置而受益。作为主要结果，我们证明了在\(\mathbb{R}^d\)中任何期望防策略机制的近似比的渐近下界为\(1 + \sqrt{d/(2(d + 1))}\)。通过为双代理情况提供一个随机的\(\sqrt{2}\)近似机制，表明这种障碍是由大量人口驱动的。然后我们考虑一个输出增强框架，对于代理被限制在一条线上但设施可在平面上任何位置的设置，我们设计了一个确定性的防策略\(\sqrt{2}\)近似机制并给出匹配下界，表明输出增强可替代随机性需求。对于代理报告位于单位圆上但设施可在\(\mathbb{R}^2\)中任何位置的设置，我们引入了一个期望下成组防策略的随机\(3/2\)近似机制。
英文摘要
We study the strategic facility location problem under the egalitarian objective, where a mechanism uses the reported locations of a set of agents in Euclidean space to select a facility location that minimizes the maximum distance to any agent. We restrict our attention to strategyproof mechanisms, ensuring that no agent can benefit from misreporting their location. As our main results, we prove an asymptotic lower bound of $1 + \sqrt{d/(2(d+1))}$ on the approximation ratio of any mechanism that is strategyproof in expectation in $\mathbb{R}^d$. We show that this barrier is driven by large populations by providing a randomized $\sqrt{2}$-approximate mechanism for the two-agent case. We then consider an output-augmented framework, which allows the facility to be placed outside the agents' restricted domain. For the setting where agents are restricted to a line but the facility can be anywhere in the plane, we design a deterministic strategyproof $\sqrt{2}$-approximate mechanism with a matching lower bound, showing that output augmentation can replace the need for randomness. For the setting where the agents' reports lie on the unit circle but the facility can be placed anywhere in $\mathbb{R}^2$ we introduce a randomized $3/2$-approximate mechanism that is group-strategyproof in expectation.
