In this paper, we investigate the Mechanism Design aspects of the $m$-Capacitated Facility Location Problem ($m$-CFLP) on a line. We focus on two frameworks. In the first framework, the number of facilities is arbitrary, all facilities have the same capacity, and the number of agents is equal to the total capacity of all facilities. In the second framework, we aim to place two facilities, each with a capacity of at least half of the total agents. For both of these frameworks, we propose truthful mechanisms with bounded approximation ratios with respect to the Social Cost (SC) and the Maximum Cost (MC). When $m>2$, the result sharply contrasts with the impossibility results known for the classic $m$-Facility Location Problem \cite{fotakis2014power}, where capacity constraints are not considered. Furthermore, all our mechanisms are (i) optimal with respect to the MC (ii) optimal or nearly optimal with respect to the SC among anonymous mechanisms. For both frameworks, we provide a lower bound on the approximation ratio that any truthful and deterministic mechanism can achieve with respect to the SC and MC.

中文翻译：

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具有容量限制的设施选址问题：两个及以上设施

在本文中，我们研究了线路上 $m$-Capacitated Facility Location Problem ($m$-CFLP) 的机制设计方面。我们关注两个框架。在第一个框架中，设施的数量是任意的，所有设施具有相同的容量，并且代理的数量等于所有设施的总容量。在第二个框架中，我们的目标是放置两个设施，每个设施的容量至少占代理总数的一半。对于这两个框架，我们提出了关于社会成本（SC）和最大成本（MC）具有有限近似比率的真实机制。当 $m>2$ 时，结果与经典 $m$-设施位置问题 \cite{fotakis2014power} 已知的不可能性结果形成鲜明对比，其中不考虑容量约束。此外，在匿名机制中，我们所有的机制都是（i）相对于 MC 最优的（ii）相对于 SC 最优或接近最优的。对于这两个框架，我们提供了任何真实且确定性机制相对于 SC 和 MC 可以实现的近似率下限。