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, collisionfree 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 nontrivial results for multiobject 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 constantfactor 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 NPhard, 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 Omega(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.
BibTeX  Entry
@InProceedings{demaine_et_al:LIPIcs:2018:8742,
author = {Erik D. Demaine and S{\'a}ndor P. Fekete and Phillip Keldenich and Christian Scheffer and Henk Meijer},
title = {{Coordinated Motion Planning: Reconfiguring a Swarm of Labeled Robots with Bounded Stretch}},
booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)},
pages = {29:129:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770668},
ISSN = {18688969},
year = {2018},
volume = {99},
editor = {Bettina Speckmann and Csaba D. T{\'o}th},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8742},
URN = {urn:nbn:de:0030drops87423},
doi = {10.4230/LIPIcs.SoCG.2018.29},
annote = {Keywords: Robot swarms, coordinated motion planning, parallel motion, makespan, bounded stretch, complexity, approximation}
}
Keywords: 

Robot swarms, coordinated motion planning, parallel motion, makespan, bounded stretch, complexity, approximation 
Collection: 

34th International Symposium on Computational Geometry (SoCG 2018) 
Issue Date: 

2018 
Date of publication: 

08.06.2018 