Obnoxious Facility Location Problems: Strategyproof Mechanisms Optimizing $L_p$-Aggregated Utilities and Costs

2025-12-21

AI Summary

This paper studies the strategic location of obnoxious facilities on the interval [0,1]: agents prefer facilities to be as far as possible from their private locations. We design (group) strategyproof mechanisms that elicit truthful location reports and approximately optimize either the Lₚ-aggregate utility (maximization) or cost (minimization) for any real p ∈ [−∞, ∞]. Our key innovation is the first unified extension of Lₚ objectives to the entire extended real line—including p = ±∞—overcoming the traditional restriction to p ∈ {1, 2, ∞}. Leveraging mechanism design, game-theoretic analysis, and generalized Lₚ-norm techniques, we establish tight approximation ratios—both upper and lower bounds—for deterministic mechanisms across all p. For randomized mechanisms, we obtain matching bounds for several p values. Our results systematically characterize the fundamental approximability limits of obnoxious facility location, significantly advancing the theoretical understanding of optimality in this domain.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the problem of locating a single obnoxious facility on the normalized line segment $[0,1]$ with strategic agents from a mechanism design perspective. Each agent has a preference for the undesirable location of the facility and would prefer the facility to be far away from their location. We consider the utility of the agent, defined as the distance between the agent's location and the facility location, and the cost of each agent, equal to one minus the utility. Given this standard setting of obnoxious facility location problems, our goal is to design (group) strategyproof mechanisms to elicit agent locations truthfully and determine facility location approximately optimizing the $L_p$-aggregated utility and cost objectives, which generalizes the $L_p$-norm ($pge 1$) of the agents' utilities and agents' costs to any $p in [-infty, infty]$, respectively. We establish upper and lower bounds on the approximation ratios of deterministic and randomized (group) strategyproof mechanisms for maximizing the $L_p$-aggregated utilities or minimizing the $L_p$-aggregated costs across the range of (p)-values. While there are gaps between upper and lower bounds for randomized mechanisms, our bounds for deterministic mechanisms are tight.

Problem

• Design strategyproof mechanisms for obnoxious facility location
• Optimize Lp-aggregated utilities and costs on a line segment
• Establish approximation bounds for deterministic and randomized mechanisms

Innovation

• Strategyproof mechanisms for obnoxious facility location
• Optimizing Lp-aggregated utilities and costs
• Tight bounds for deterministic mechanisms