 
Summary: The FreezeTag Problem: How to Wake Up a Swarm of Robots #
Esther M. Arkin + Michael A. Bender # S’andor P. Fekete § Joseph S. B. Mitchell +
Martin Skutella ¶
Abstract
An optimization problem that naturally arises in the study of swarm robotics is the FreezeTag
Problem (FTP) of how to awaken a set of ``asleep'' robots, by having an awakened robot move to their
locations. Once a robot is awake, it can assist in awakening other slumbering robots. The objective
is to have all robots awake as early as possible. While the FTP bears some resemblance to problems
from areas in combinatorial optimization such as routing, broadcasting, scheduling, and covering, its
algorithmic characteristics are surprisingly di#erent.
We consider both scenarios on graphs and in geometric environments. In graphs, robots sleep at
vertices and there is a length function on the edges. Awake robots travel along edges, with time depending
on edge length. For most scenarios, we consider the o#ine version of the problem, in which each awake
robot knows the position of all other robots. We prove that the problem is NPhard, even for the special
case of star graphs. We also establish hardness of approximation, showing that it is NPhard to obtain
an approximation factor better than 5/3, even for graphs of bounded degree.
These lower bounds are complemented with several positive algorithmic results, including:
. We show that the natural greedy strategy on star graphs has a tight worstcase performance of 7/3
and give a polynomialtime approximation scheme (PTAS) for star graphs.
. We give a simple O(log #)competitive online algorithm for graphs with maximum degree # and
