  We present a simple and natural extension of the multi-robot motion planning
problem where the robots are partitioned into groups (colors), such that in
each group the robots are interchangeable. Every robot is no longer required to
move to a specific target, but rather to some target placement that is assigned
to its group. We call this problem k-color multi-robot motion planning and
provide a sampling-based algorithm specifically designed for solving it. At the
heart of the algorithm is a novel technique where the k-color problem is
reduced to several discrete multi-robot motion planning problems. These
reductions amplify basic samples into massive collections of free placements
and paths for the robots. We demonstrate the performance of the algorithm by an
implementation for the case of disc robots and polygonal robots translating in
the plane. We show that the algorithm successfully and efficiently copes with a
variety of challenging scenarios, involving many robots, while a simplified
version of this algorithm, that can be viewed as an extension of a prevalent
sampling-based algorithm for the k-color case, fails even on simple scenarios.
Interestingly, our algorithm outperforms a well established implementation of
PRM for the standard multi-robot problem, in which each robot has a distinct
color.
