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.

翻译：本文研究了线上$m$容量约束设施选址问题（$m$-CFLP）的机制设计方面。我们聚焦于两个框架。在第一个框架中，设施数量任意，所有设施具有相同容量，且代理人总数等于所有设施的总容量。在第二个框架中，我们旨在放置两个设施，每个设施的容量至少为代理人总数的一半。针对这两个框架，我们提出了关于社会成本（SC）和最大成本（MC）具有有界近似比的无偏机制。当$m>2$时，该结果与经典$m$设施选址问题中已知的不可能性结果形成鲜明对比\cite{fotakis2014power}，后者未考虑容量约束。此外，我们所有的机制在（i）关于MC是最优的，（ii）在匿名机制中关于SC是最优或接近最优的。对于这两个框架，我们给出了任何真实且确定性机制在SC和MC方面所能达到的近似比的下界。