Geometric Freeze-Tag Problem

We study the Freeze-Tag Problem (FTP), introduced by Arkin et al. (SODA'02), where the goal is to wake up a group of $n$ robots, starting from a single active robot. Our focus is on the geometric version of the problem, where robots are positioned in $\mathbb{R}^d$, and once activated, a robot can move at a constant speed to wake up others. The objective is to minimize the time it takes to activate the last robot, also known as the makespan.We present new upper bounds for the $l_1$ and $l_2$ norms in $\mathbb{R}^2$ and $\mathbb{R}^3$. For $(\mathbb{R}^2, l_2)$, we achieve a makespan of at most $5.4162r$, improving on the previous bound of $7.07r$ by Bonichon et al. (DISC'24). In $(\mathbb{R}^3, l_1)$, we establish an upper bound of $13r$, which leads to a bound of $22.52r$ for $(\mathbb{R}^3, l_2)$. Here, $r$ denotes the maximum distance of a robot from the initially active robot under the given norm. To the best of our knowledge, these are the first known bounds for the makespan in $\mathbb{R}^3$ under these norms.We also explore the FTP in $(\mathbb{R}^3, l_2)$ for specific instances where robots are positioned on a boundary, providing further insights into practical scenarios.