Constant-Approximate and Constant-Strategyproof Two-Facility Location

TL;DR

This paper introduces new mechanisms for the two-facility location problem that are both approximately optimal and strategyproof under relaxed conditions, including allowing facilities in the plane and ensuring Lipschitz continuity.

Contribution

It presents the first constant-approximate, constant-strategyproof mechanisms for two-facility location with relaxed assumptions, including planar facility placement and Lipschitz continuity.

Findings

A natural mechanism achieves constant approximation and strategyproofness.

A Lipschitz continuous mechanism maintains constant approximation and strategyproofness.

Mechanisms exhibit stability properties despite large input changes.

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 $\Omega(n)$-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 $\Omega(n)$ 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.