Distributed Agent-Constrained Truthful Facility Location

TL;DR

This work analyzes a distributed facility location problem on the real line with agents partitioned into fixed groups, where $k$ facilities are opened at positions reported by agents via a two-phase process: each group selects a representative and then $k$ representatives are chosen as facility locations. It develops deterministic strategyproof mechanisms and derives tight approximation bounds for the social cost in the two-facility case: $1+\sqrt{2}$ for the sum-variant and $\tfrac{9}{2}$ for the max-variant, with matching lower bounds; for unrestricted algorithms, the sum-variant bound remains $1+\sqrt{2}$ while the max-variant bound is $4$. For $k\ge 3$ facilities, the paper proves that strategyproof mechanisms can achieve approximation ratios between $3-\tfrac{2}{k}$ and $3+\tfrac{2}{k}$ (sum-variant) and between $2k$ and $2(k+1)$ (max-variant), and introduces generalized mechanisms $\mathcal{M}_k^{\text{sum}}$ and $\mathcal{M}_k^{\text{max}}$ based on ordered statistics to attain these bounds. The results illuminate the trade-offs between truthfulness and efficiency in distributed decision-making and point to future directions including randomized mechanisms and alternative social objectives.

Abstract

We study a distributed facility location problem in which a set of agents, each with a private position on the real line, is partitioned into a collection of fixed, disjoint groups. The goal is to open $k$ facilities at locations chosen from the set of positions reported by the agents. This decision is made by mechanisms that operate in two phases. In Phase 1, each group selects the position of one of its agents to serve as the group's representative location. In Phase 2, $k$ representatives are chosen as facility locations. Once the facility locations are determined, each agent incurs an individual cost, defined either as the sum of its distances to all facilities (sum-variant) or as the distance to its farthest facility (max-variant). We focus on the class of strategyproof mechanisms, which preclude the agents from benefiting through strategic misreporting, and establish tight bounds on the approximation ratio with respect to the social cost (the total individual agent cost) in both variants.

Theorems & Definitions (28)

• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof