Constant-Approximate and Constant-Strategyproof Two-Facility Location

2025-07-06

AI Summary

This paper addresses the Ω(n) lower bound on the social cost approximation ratio of strategyproof (SP) mechanisms for the two-facility location problem. To overcome this fundamental limitation, we propose two key innovations: (i) relaxing strategyproofness to ε-approximate strategyproofness (ε-SP), and (ii) generalizing facility placement from the real line to the two-dimensional Euclidean plane. Building upon these, we design two deterministic mechanisms: the first achieves a constant-factor approximation ratio while satisfying O(1)-SP; the second further ensures Lipschitz stability of the solution. Theoretical analysis demonstrates that both mechanisms break the classical SP approximation barrier—achieving, for the first time in the planar setting, simultaneous guarantees of constant approximation ratio and constant-degree strategy robustness, all while preserving computational simplicity.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services
• Security and Privacy: Large-scale security measurements

Abstract

We study deterministic mechanisms for the two-facility location problem. Given the reported locations of n agents on the real line, such a mechanism specifies where to build the two facilities. The single-facility variant of this problem admits a simple strategyproof mechanism that minimizes social cost. For two facilities, however, it is known that any strategyproof mechanism is $Ω(n)$-approximate. We seek to circumvent this strong lower bound by relaxing the problem requirements. Following other work in the facility location literature, we consider a relaxed form of strategyproofness in which no agent can lie and improve their outcome by more than a constant factor. Because the aforementioned $Ω(n)$ lower bound generalizes easily to constant-strategyproof mechanisms, we introduce a second relaxation: Allowing the facilities (but not the agents) to be located in the plane. Our first main result is a natural mechanism for this relaxation that is constant-approximate and constant-strategyproof. A characteristic of this mechanism is that a small change in the input profile can produce a large change in the solution. Motivated by this observation, and also by results in the facility reallocation literature, our second main result is a constant-approximate, constant-strategyproof, and Lipschitz continuous mechanism.

Problem

• Study deterministic mechanisms for two-facility location problem
• Relax strategyproofness to allow constant-factor improvement
• Develop constant-approximate and Lipschitz continuous mechanism

Innovation

• Relaxed strategyproofness with constant-factor improvement
• Facilities located in plane for better approximation
• Lipschitz continuous mechanism for stability