Facility Location Problems with Capacity Constraints: Two Facilities and Beyond

TL;DR

The paper addresses mechanism design for the $m$-CFLP on a line under two regimes: equi-capacitated facilities with no spare capacity and two abundant facilities with possibly different capacities. It introduces two truthful anonymous mechanisms, PMM and PIPM, for the equi-capacitated setting, establishing finite approximation ratios for both Social and Maximum Costs and proving tight lower bounds that render these mechanisms optimal for MC and near-optimal for SC under anonymity constraints. For the two-facility abundant-capacity regime, it proposes the Extended InnerGap (EIG) mechanism, proving strong Group Strategyproofness and bounded approximations, with MC optimal and SC near-optimal under conditions on the total capacity and number of agents. Overall, the work demonstrates that capacitated facility location on a line can circumvent classic impossibility results for uncapacitated FLP and provides foundational lower bounds to guide future mechanism design in constrained settings.

Abstract

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.

Theorems & Definitions (38)

• Theorem 1
• proof
• Example 1
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• Theorem 5
• proof
• Theorem 6