Facility Location Problems with Capacity Constraints: Two Facilities and Beyond

TL;DR

This paper studies capacity-constrained facility location problems on a line, proposing truthful mechanisms with bounded approximation ratios for two frameworks, and contrasting results with classic unconstrained problems.

Contribution

It introduces new truthful mechanisms for capacity-constrained facility location on a line, achieving optimal or near-optimal approximation ratios, and provides lower bounds for these problems.

Findings

Mechanisms are optimal for maximum cost.

Mechanisms are nearly optimal for social cost among anonymous mechanisms.

Contrasts with classic unconstrained facility location results.

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.