We consider the distributed facility location problem, in which there is a set of agents positioned on the real line, who are also partitioned into multiple symmetric districts. The goal is to choose a single location (where a public facility is to be built) so as to minimize the total distance of the agents from that location. Importantly, this process is distributed: the positions of the agents in each district are first aggregated into a representative location for the district, and then one of the representatives is chosen as the facility location. This indirect access to the positions of the agents inevitably leads to inefficiency, which is captured by the notion of distortion. We study both the discrete version of the problem, where the set of alternative locations if finite, as well as the continuous one, where every point of the line is an alternative. For both versions, we paint an almost complete picture of the distortion landscape of distributed mechanisms. We start from the discrete setting, for which we show a tight bound of 3 on the distortion of general mechanisms and a tight bound of 7 for strategyproof mechanisms. For the continuous setting, we show that the distortion of general mechanisms lies between 2 and 3, whereas the distortion of strategyproof mechanisms is exactly 3.

中文翻译：

---

分布式设施选址的近似机制设计

我们考虑分布式设施定位问题，其中有一组位于实线上的代理，他们也被划分为多个对称区域。目标是选择一个位置（将要建造公共设施的位置）以最小化代理与该位置的总距离。重要的是，这个过程是分布式的：首先将每个区的代理位置聚合到该区的一个代表位置，然后选择其中一个代表作为设施位置。这种对代理位置的间接访问不可避免地导致效率低下，这被扭曲的概念所捕获。我们研究了问题的离散版本，其中替代位置的集合如果有限，以及连续的，这条线的每个点都是一个选择。对于这两个版本，我们几乎完整地描绘了分布式机制的失真情况。我们从离散设置开始，我们在一般机制的失真上显示了 3 的严格界限，对于策略证明机制的严格界限为 7。对于连续设置，我们表明一般机制的失真介于 2 和 3 之间，而策略证明机制的失真正好是 3。