Freeze-Tag: Optimal Strategies for Awakening a Swarm of Robots

An optimization problem that naturally arises in the study of \textquotedblleft{}swarm robotics\textquotedblright{} is to wake up a set of \textquotedblleft{}asleep\textquotedblright{} robots, starting with only one \textquotedblleft{}awake\textquotedblright{} robot. One robot can only awaken another when they are in the same location. As soon as a robot is awake, it may assist in waking up other robots. The goal is to compute an optimal awakening schedule such that all robots are awake by time $t^*$, for the smallest possible value of $t^*$ (the optimal makespan). The $n$ robots are initially at $n$ points of a metric space. The problem is equivalent to finding a spanning tree with maximum out-degree two that minimizes the radius from a fixed source.

Is it NP-hard to determine an optimal awakening schedule for robots in the Euclidean (or $L_1$) plane? In more general metric spaces, can one obtain an approximation algorithm with better than $O(\log n)$ performance ratio?