Freeze-Tag is NP-hard in 2D with $L_1$ distance

Abstract

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.