  For a multi-robot team that collaboratively explores an unknown environment,
it is of vital importance that collected information is efficiently shared
among robots in order to support exploration and navigation tasks. Practical
constraints of wireless channels, such as limited bandwidth, urge robots to
carefully select information to be transmitted. In this paper, we consider the
case where environmental information is modeled using a 3D Scene Graph, a
hierarchical map representation that describes both geometric and semantic
aspects of the environment. Then, we leverage graph-theoretic tools, namely
graph spanners, to design greedy algorithms that efficiently compress 3D Scene
Graphs with the aim of enabling communication between robots under bandwidth
constraints. Our compression algorithms are navigation-oriented in that they
are designed to approximately preserve shortest paths between locations of
interest, while meeting a user-specified communication budget constraint. The
effectiveness of the proposed algorithms is demonstrated in synthetic robot
navigation experiments in a realistic simulator. A video abstract is available
at https://youtu.be/nKYXU5VC6A8.
