Constrained Truthful Obnoxious Two-Facility Location with Optional Preferences

Abstract

We consider a truthful facility location problem with agents that have private positions on the line of real numbers and known optional preferences over two obnoxious facilities that must be placed at locations chosen from a given set of candidate ones. Each agent wants to be as far away as possible from the facilities that affect her, and our goal is to design mechanisms that decide where to place the facilities so as to maximize the total happiness of the agents as well as provide the right incentives to them to truthfully report their positions. We consider separately the setting in which all agents are affected by both facilities (i.e., they have non-optional preferences) and the general optional setting. We show tight bounds on the approximation ratio of deterministic strategyproof mechanisms for both settings, and almost tight bounds for randomized mechanisms.

Figures (2)

• Figure 1: The instances considered in the proof of Theorem \ref{['thm:doubleton:lower:deterministic']} when the solution chosen for instance $I$ is $(2,2)$. Instance $J$ is obtained by moving the $(1-\alpha)n$ agents at $1+\varepsilon$ in $I$ to $2$. Instance $Q$ is obtained by moving the $\alpha n$ agents at $0$ in $J$ to $1-\varepsilon$. In all these instances, the mechanism must choose solution $(2,2)$ due to strategyproofness, which leads to the lower bound of $\sqrt{3}$ on its approximation ratio.
• Figure 2: The instances considered in the proof of Theorem \ref{['thm:doubleton:lower:randomized']}.

Theorems & Definitions (19)

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