In the standard Freeze-Tag Problem (FTP), an initially awake robot (the source) is in charge of waking up a swarm of sleeping robots by moving towards them, given that all the awake robots can participate in the awakening process. The goal is to minimize the makespan to wake up all robots assuming they move at unit speed. In this paper we introduce the Freeze-Tag-with-Return Problem (FTRP) variant, where the robots must eventually return to their initial positions. In the Euclidean plane with n sleeping robots lying on the unit disk centered at the initial position of the source, we show a non-trivial relationship between FTP and FTRP by proving that the difference between the optimal makespan of both problems never exceeds 1.959, and is at least 1.732 in the worst-case. We also present several upper and lower bounds on the optimal makespan. In particular, we show that if the sleeping robots are in convex positions, then the optimal makespan is at most 2 + 2√2, which is achieved by some instances. From an algorithmic point-of-view, we present single-exponential algorithms for general distance functions. In metric spaces, these algorithms are asymptotically optimal under the ETH, which we show via an NP-hardness reduction on unweighted graphs.

The Freeze-Tag Problem consists in waking up a swarm of robots starting with one initially awake robot. While there exists a wide literature on the centralized setting, where the locations of the robots are known in advance, we focus on the distributed version where the locations of the robots, P, are unknown, and where awake robots only detect other robots up to distance 1. Assuming that moving at distance δ takes a time δ, we show that waking up the whole swarm takes O(ρ + ℓ2 log(ρ/ℓ)), where ρ is the largest distance from the initial robot to any point of P, and ℓ is the connectivity threshold of P. Moreover, the result is complemented by a matching lower bound. We also provide other distributed algorithms, complemented with lower bounds, whenever each robot has a bounded amount of energy.