  We study the problem of optimally inspecting an underground (underwater)
gallery with k agents. We consider a gallery with a single opening and with a
tree topology rooted at the opening. Due to the small diameter of the pipes
(caves), the agents are small robots with limited autonomy and there is a
supply station at the gallery's opening. Therefore, they are initially placed
at the root and periodically need to return to the supply station. Our goal is
to design off-line strategies to efficiently cover the tree with $k$ small
robots. We consider two objective functions: the covering time (maximum
collective time) and the covering distance (total traveled distance). The
maximum collective time is the maximum time spent by a robot needs to finish
its assigned task (assuming that all the robots start at the same time); the
total traveled distance is the sum of the lengths of all the covering walks.
Since the problems are intractable for big trees, we propose approximation
algorithms. Both efficiency and accuracy of the suboptimal solutions are
empirically showed for random trees through intensive numerical experiments.
