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

TL;DR

This work resolves the open question of FTP complexity in the plane under the $L_1$ distance by proving strong NP-hardness via a reduction from Numerical $3$-Dimensional Matching, constructing a carefully arranged FTP instance with groups $A,A',B,B',C$ and parameters $L$, $\varepsilon$, and $\delta$ so that a schedule with makespan $L$ corresponds to a solution of the N3DM instance. The reduction shows that every feasible FTP schedule encodes a valid triple assignment and, conversely, a valid N3DM solution yields a feasible FTP wake-up strategy, with distances computed in the $L_1$ metric guiding the necessary temporal separations. As a corollary, FTP is NP-complete in unweighted grid graphs, and the results settle a long-standing conjecture (Arkin et al., TOPP Problem 35) about the hardness of Manhattan-distance FTP in the plane. The paper also highlights open directions for constant-factor approximations on weighted graphs or trees.

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.

Figures (1)

• Figure 1: Non-scale example of Theorem \ref{['teo-1']} reduction.

Theorems & Definitions (5)

• Conjecture 1: Arkin et al. Arkin06
• Theorem 1
• proof
• Corollary 1
• Corollary 2