Freeze-Tag is NP-Hard in 3D with $L_1$ distance

TL;DR

This paper proves that the Freeze-Tag Problem, involving activating robots in 3D space with L1 distance, is NP-hard, filling a key gap in understanding its computational complexity.

Contribution

It establishes NP-hardness of the Freeze-Tag Problem in 3D with L1 distance, extending previous results from Euclidean and other Lp spaces.

Findings

FTP is NP-hard in 3D with L1 distance

Completes the complexity classification in 3D spaces

Bridges the gap between Euclidean and L1 cases

Abstract

Arkin et al. in 2002 introduced a scheduling-like problem called Freeze-Tag Problem (FTP) motivated by robot swarm activation. The input consists of the locations of n mobile punctual robots in some metric space or graph. Only one begins "active", while the others are initially "frozen". All active robots can move at unit speed and, upon reaching a frozen one's location, activates it. The goal is to activate all the robots in the minimum amount of time, the so-called makespan. Until 2017 the hardness of this problem in metric spaces was still open, but then Yu et al. proved it to be NP-Hard in the Euclidian plane, and in the same year, Demaine and Roudoy demonstrated that the FTP is also hard in 3D with any $L_p$ distance (with p > 1). However, we still don't know whether Demaine's and Roudoy's result could be translated to the plane. This paper fills the p=1 gap by showing that the FTP is NP-Hard in 3D with $L_1$ distance.