The Freeze-Tag Problem (FTP) is a scheduling problem with application in robot swarm activation and was introduced by Arkin et al. in 2002. This problem seeks an efficient way of activating a robot swarm, starting with a single active robot. Activations occur through direct contact, and once a robot becomes active, it can move and help activate other robots. Although the problem has been shown to be NP-hard in the Euclidean plane $\mathbb{R}^2$ under the $L_2$ distance, and in three-dimensional Euclidean space $\mathbb{R}^3$ under any $L_p$ distance with $p \ge 1$, its complexity under the $L_1$ (Manhattan) distance in $\mathbb{R}^2$ has remained an open question. In this paper, we settle this question by proving that FTP is strongly NP-hard in the Euclidean plane with $L_1$ distance.

翻译：冻结标记问题（FTP）是由Arkin等人在2002年提出的一个调度问题，应用于机器人群体激活。该问题旨在寻找一种从单个活跃机器人开始、高效激活整个机器人群体的方法。激活通过直接接触发生，一旦机器人被激活，它便可以移动并协助激活其他机器人。尽管该问题在$L_2$距离下的欧几里得平面$\mathbb{R}^2$中，以及在任意$p \ge 1$的$L_p$距离下的三维欧几里得空间$\mathbb{R}^3$中已被证明是NP难的，但其在$\mathbb{R}^2$中$L_1$（曼哈顿）距离下的复杂度一直是一个悬而未决的问题。本文通过证明FTP在$L_1$距离下的欧几里得平面中是强NP难的，解决了这一问题。