Truthful Two-Facility Location with Candidate Locations

TL;DR

This work studies truthful two-facility location with candidate locations on a line, where agents have private positions and public approvals over two facilities, and each agent's cost is the sum of distances to approved facilities. The authors design deterministic strategyproof mechanisms to place two facilities from a finite candidate set, achieving constant-approximation guarantees for both the social cost and max cost across doubleton, singleton, and general preference classes, including when facilities may coincide. Key mechanisms include Median-based variants, Stronger-Majority-Median, Vote-for-Priority, Leftmost, and $\alpha$-Statistic, with carefully tailored analyses that yield tight or near-tight bounds such as $1+\sqrt{2}$, $3$, $7$, and $2$–$3$ across settings. The results advance understanding of distortion under truthfulness in discrete two-facility locations and highlight fundamental trade-offs between incentive compatibility and efficiency, with implications for public-location planning and mechanism design without monetary incentives.

Abstract

We study a truthful two-facility location problem in which a set of agents have private positions on the line of real numbers and known approval preferences over two different facilities. Given the locations of the two facilities, the cost of an agent is the total distance from the facilities she approves. The goal is to decide where to place the facilities from a given finite set of candidate locations so as to (a) approximately optimize desired social objectives, and (b) incentivize the agents to truthfully report their private positions. We focus on the class of deterministic strategyproof mechanisms and show bounds on their approximation ratio in terms of the social cost (i.e., the total cost of the agents) and the max cost for several classes of instances depending on the preferences of the agents over the facilities.

Figures (4)

• Figure 1: The instances used in the proof of the lower bound of $1+\sqrt{2}$ in terms of the social cost for doubleton instances (Theorem \ref{['thm:sc-doubleton-lower']}). Set $A$ consists of $\alpha n$ agents and set $B$ consists of $(1-\alpha)n$ agents; all of them approve both facilities. Rectangles represent candidate locations; recall that we assume that there are two candidate locations arbitrarily close to $0$ and two candidate locations arbitrarily close to $2$.
• Figure 2: The two instances used in the proof of the lower bound of $3$ in terms of the social cost for the general case (Theorem \ref{['thm:sc-general-lower']}). Agent $i$ approves $F_1$ and agent $j$ approves $F_2$. Rectangles represent candidate locations.
• Figure 3: The two instances used in the proof of the lower bound of $2$ in terms of the max cost for doubleton instances (Theorem \ref{['max:both-lower-2']}). Both agents $i$ and $j$ approve both facilities. Rectangles represent candidate locations.
• Figure 4: The two instances used in the proof of the lower bound of $3$ in terms of the max cost for singleton instances (Theorem \ref{['max:singleton-lower-3']}). Both agents $i$ and $j$ approve facility $F_1$. Rectangles represent candidate locations.

Theorems & Definitions (40)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Example 3.4
• Theorem 3.5
• proof
• Theorem 3.6