  An optimization problem that naturally arises in the study of swarm robotics
is the Freeze-Tag 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
different. 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 offline version of the problem, in
which each awake robot knows the position of all other robots. We prove that
the problem is NP-hard, even for the special case of star graphs. We also
establish hardness of approximation, showing that it is NP-hard 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: (1) We show that the natural greedy strategy on star graphs has a
tight worst-case performance of 7/3 and give a polynomial-time approximation
scheme (PTAS) for star graphs. (2) We give a simple O(log D)-competitive online
algorithm for graphs with maximum degree D and locally bounded edge weights.
(3) We give a PTAS, running in nearly linear time, for geometrically embedded
instances.
