**Reference**:- Erik D. Demaine, Sándor P. Fekete, Phillip Keldenich, Christian Scheffer, and Henk Meijer, “Coordinated Motion Planning: Reconfiguring a Swarm of Labeled Robots with Bounded Stretch”, in
*Proceedings of the 34th International Symposium on Computational Geometry (SoCG 2018)*, June 11–14, 2018, 29:1–29:15. **Abstract**:-
We present a number of breakthroughs for coordinated motion planning, in which
the objective is to reconfigure a swarm of labeled convex objects by a
combination of parallel, continuous, collision-free translations into a given
target arrangement. Problems of this type can be traced back to the classic
work of Schwartz and Sharir (1983), who gave a method for deciding the
existence of a coordinated motion for a set of disks between obstacles; their
approach is polynomial in the complexity of the obstacles, but exponential in
the number of disks. Despite a broad range of other non-trivial results for
multi-object motion planning, previous work has largely focused on
*sequential*schedules, in which one robot moves at a time, with objectives such as the number of moves; attempts to minimize the overall makespan of a coordinated*parallel*motion schedule (with many robots moving simultaneously) have defied all attempts at establishing the complexity in the absence of obstacles, as well as the existence of efficient approximation methods.We resolve these open problems by developing a framework that provides constant-factor approximation algorithms for minimizing the execution time of a coordinated,

*parallel*motion plan for a swarm of robots in the absence of obstacles, provided their arrangement entails some amount of separability. In fact, our algorithm achieves constant stretch factor: If all robots want to move at most*d*units from their respective starting positions, then the total duration of the overall schedule (and hence the distance traveled by each robot) is*O*(*d*). Various extensions include unlabeled robots and different classes of robots. We also resolve the complexity of finding a reconfiguration plan with minimal execution time by proving that this is NP-hard, even for a grid arrangement without any stationary obstacles. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor Ω(*N*^{1/4}) may be required. On the positive side, we establish a stretch factor of*O*(*N*^{1/2}) even in this case. The intricate difficulties of computing precise optimal solutions are demonstrated by the seemingly simple case of just two disks, which is shown to be excruciatingly difficult to solve to optimality. **Length**:- The paper is 15 pages.
**Availability**:- The paper is available in PDF (576k).
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Last updated December 1, 2021 by Erik Demaine.