  We study the problem of multi-robot target assignment to minimize the total
distance traveled by the robots until they all reach an equal number of static
targets. In the first half of the paper, we present a necessary and sufficient
condition under which true distance optimality can be achieved for robots with
limited communication and target-sensing ranges. Moreover, we provide an
explicit, non-asymptotic formula for computing the number of robots needed to
achieve distance optimality in terms of the robots' communication and
target-sensing ranges with arbitrary guaranteed probabilities. The same bounds
are also shown to be asymptotically tight.
  In the second half of the paper, we present suboptimal strategies for use
when the number of robots cannot be chosen freely. Assuming first that all
targets are known to all robots, we employ a hierarchical communication model
in which robots communicate only with other robots in the same partitioned
region. This hierarchical communication model leads to constant approximations
of true distance-optimal solutions under mild assumptions. We then revisit the
limited communication and sensing models. By combining simple rendezvous-based
strategies with a hierarchical communication model, we obtain decentralized
hierarchical strategies that achieve constant approximation ratios with respect
to true distance optimality. Results of simulation show that the approximation
ratio is as low as 1.4.
