Freeze-Tag with Return

2026-06-20

AI Summary

This work presents the first systematic study of the Freeze-Tag problem with Return constraints (FTRP), where all activated robots must ultimately return to the starting point, and the objective is to minimize the makespan. Building upon the classical Freeze-Tag problem, the authors introduce the return requirement and prove that FTRP is NP-hard in metric spaces. Under the Exponential Time Hypothesis (ETH), they propose a single-exponential time algorithm that is asymptotically optimal for general distance functions. For robots confined to the unit disk in the Euclidean plane or placed in convex position, they establish bounds on the gap between the optimal makespans of FTRP and the original problem: a lower bound of √3 ≈ 1.732 and an upper bound of 1.959. Moreover, in the convex setting, they derive a tight upper bound of 2 + 2√2.

Abstract

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 $2 + 2\sqrt{2}$, 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.

Problem

• Freeze-Tag
• makespan
• robot awakening
• return constraint
• optimization

Innovation

• Freeze-Tag-with-Return
• makespan bounds
• single-exponential algorithm
• convex configuration
• ETH-hardness