On the design of truthful mechanisms for the capacitated facility location problem with two and more facilities

In this paper, we explore the Mechanism Design aspects of the m-Capacitated Facility Location Problem (m-CFLP) on a line, focusing on two frameworks. In the first framework, the number of facilities is arbitrary, all facilities share the same capacity, and the number of agents matches the total capacity of the facilities. In the second framework, we need to locate two facilities, each with a capacity equal to at least half the number of agents. For both frameworks, we propose truthful mechanisms with bounded approximation ratios in terms of Social Cost (SC) and Maximum Cost (MC). When m > 2, our results stand in contrast to the impossibility results known for the classical m-Facility Location Problem, where capacity constraints are absent. Moreover, all the proposed mechanisms are optimal with respect to MC and either optimal or near-optimal with respect to the SC among anonymous mechanisms. We then establish lower bounds on the approximation ratios that any truthful and deterministic mechanism achieves with respect to SC and MC for both frameworks. Lastly, we run several numerical experiments to empirically evaluate the performances of our mechanisms with respect to the SC or the MC. Our empirical analysis shows that our proposed mechanisms outperform all previously proposed mechanisms applicable in this setting.