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

TL;DR

This paper designs strategyproof mechanisms for locating an obnoxious facility on a line, optimizing generalized $L_p$-aggregated utilities and costs, with tight bounds for deterministic mechanisms and bounds for randomized ones.

Contribution

It introduces a unified framework for $L_p$-aggregation in obnoxious facility location, providing tight bounds for deterministic mechanisms and bounds for randomized mechanisms.

Findings

Tight bounds established for deterministic mechanisms.

Upper and lower bounds for randomized mechanisms.

Generalization of $L_p$-aggregation for utility and cost objectives.

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 ($p\ge 1$) 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.