Optimal randomized truthful facility location in the plane

Determine the optimal approximation ratio achievable by randomized truthful mechanisms for the egalitarian facility location problem in the two-dimensional Euclidean space when both agents and the facility are constrained to lie in the plane.

Background

The paper studies strategyproof facility location under the egalitarian objective, which minimizes the expected maximum distance between the facility and any agent. The optimal randomized approximation ratio is known to be 1.5 when agents and the facility are restricted to the real line, but the corresponding problem in the two-dimensional Euclidean plane is not fully resolved for general numbers of agents.

The paper resolves the special case of two agents in the plane by constructing a truthful mechanism with approximation ratio √2 and proving its optimality. However, the general planar problem for an arbitrary number of agents remains the explicitly identified unresolved question.