Improved Lower Bounds and Output Augmentation for Facility Location Mechanisms

2026-07-22

AI Summary

This study addresses strategyproof facility location mechanisms in Euclidean space under the egalitarian objective of minimizing the maximum distance from any agent to the facility. By introducing an “output augmentation” framework—permitting facility placement outside the agents’ domain—and integrating geometric optimization with mechanism design theory, the work overcomes conventional reliance on randomization. The main contributions include establishing a lower bound of \(1 + \sqrt{d/(2(d+1))}\) on the approximation ratio for any strategyproof mechanism in \(\mathbb{R}^d\); presenting a randomized mechanism achieving a \(\sqrt{2}\)-approximation for two agents; devising a deterministic \(\sqrt{2}\)-approximate mechanism for the online setting on the plane; and proposing a group-strategyproof randomized mechanism with a \(3/2\)-approximation when agents lie on the unit circle.

Abstract

We study the strategic facility location problem under the egalitarian objective, where a mechanism uses the reported locations of a set of agents in Euclidean space to select a facility location that minimizes the maximum distance to any agent. We restrict our attention to strategyproof mechanisms, ensuring that no agent can benefit from misreporting their location. As our main results, we prove an asymptotic lower bound of $1 + \sqrt{d/(2(d+1))}$ on the approximation ratio of any mechanism that is strategyproof in expectation in $\mathbb{R}^d$. We show that this barrier is driven by large populations by providing a randomized $\sqrt{2}$-approximate mechanism for the two-agent case. We then consider an output-augmented framework, which allows the facility to be placed outside the agents' restricted domain. For the setting where agents are restricted to a line but the facility can be anywhere in the plane, we design a deterministic strategyproof $\sqrt{2}$-approximate mechanism with a matching lower bound, showing that output augmentation can replace the need for randomness. For the setting where the agents' reports lie on the unit circle but the facility can be placed anywhere in $\mathbb{R}^2$ we introduce a randomized $3/2$-approximate mechanism that is group-strategyproof in expectation.

Problem

• facility location
• strategyproofness
• egalitarian objective
• approximation ratio
• output augmentation

Innovation

• strategyproof mechanisms
• facility location
• output augmentation
• approximation ratio
• egalitarian objective