When considering motion planning for a swarm of n labeled robots, we need
to rearrange a given start configuration into a desired target configuration
via a sequence of parallel, continuous, collision-free robot motions. The
objective is to reach the new configuration in a minimum amount of time; an
important constraint is to keep the swarm connected at all times. Problems of
this type have been considered before, with recent notable results achieving
constant stretch for not necessarily connected reconfiguration: If mapping the
start configuration to the target configuration requires a maximum Manhattan
distance of d, the total duration of an overall schedule can be bounded to
O(d), which is optimal up to constant factors. However, constant
stretch could only be achieved if disconnected reconfiguration is allowed, or
for scaled configurations (which arise by increasing all dimensions of a given
object by the same multiplicative factor) of unlabeled robots.
We resolve these major open problems by (1) establishing a lower bound of
Ω(n) for connected, labeled reconfiguration and, most
importantly, by (2) proving that for scaled arrangements, constant stretch for
connected reconfiguration can be achieved. In addition, we show that (3) it is
NP-hard to decide whether a makespan of 2 can be achieved, while it is possible
to check in polynomial time whether a makespan of 1 can be achieved.Comment: 26 pages, 17 figures, full version of an extended abstract accepted
for publication in the proceedings of the 33rd International Symposium on
Algorithms and Computation (ISAAC 2022