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

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.