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# Problem 35: Freeze-Tag: Optimal Strategies for Awakening a Swarm of Robots

Statement
An optimization problem that naturally arises in the study of ``swarm robotics'' is to wake up a set of ``asleep'' robots, starting with only one ``awake'' robot. One robot can only awaken another when they are in the same location. As soon as a robot is awake, it may assist in waking up other robots. The goal is to compute an optimal awakening schedule such that all robots are awake by time t*, for the smallest possible value of t* (the optimal makespan). The n robots are initially at n points of a metric space. The problem is equivalent to finding a spanning tree with maximum out-degree two that minimizes the radius from a fixed source.

Is it NP-hard to determine an optimal awakening schedule for robots in the Euclidean (or L1) plane? In more general metric spaces, can one obtain an approximation algorithm with better than O(log n) performance ratio?

Origin
[ABF+02]
Status/Conjectures
[ABF+02] conjecture that the freeze-tag problem is NP-hard in the Euclidean (or L1) plane. (They show it to be NP-complete in star metrics.)
Motivation
What is the most efficient way to "turn on" a large swarm of robots or to distribute to them a secret or a token that requires close proximity in order to pass from one to another?
Partial and Related Results
There are a variety of related results given in [ABF+02]. They show that the problem is NP-hard for "star metrics" (each asleep robot is at a leaf of a star graph whose spokes have various lengths). For geometric instances (Lp metrics) in fixed dimension, they give an efficient PTAS. For general metric spaces, they give an O(log n)-approximation algorithm. They also give improved approximation methods for other special cases (star graphs, ultrametrics)
Appearances
Posed in [ABF+02], and by Joseph Mitchell during the open problem session at the Fall Workshop on Computational Geometry, Brooklyn, NY, Nov. 2-3, 2001.
Categories
optimization; scheduling; robotics
Entry Revision History
E. Demaine, 20 Nov. 2001; J. Mitchell, 21 Nov. 2001.

## Bibliography

ABF+02
E. M. Arkin, M. A. Bender, S. P. Fekete, J. S. B. Mitchell, and M. Skutella.
The freeze-tag problem: How to wake up a swarm of robots.
In Proc. 13th ACM-SIAM Sympos. Discrete Algorithms, 2002.
To appear.

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The Open Problems Project - September 19, 2017