Coordinated Motion Planning: Reconfiguring a Swarm of Labeled Robots with Bounded Stretch

TL;DR

This paper introduces constant-factor approximation algorithms for parallel motion planning of labeled convex robots, achieving near-optimal reconfiguration times under certain separability conditions, and analyzes complexity and bounds for various scenarios.

Contribution

It provides the first constant-factor approximation algorithms for parallel reconfiguration of labeled robots and establishes complexity and stretch bounds for dense packings.

Findings

Achieves constant stretch factor for reconfiguration without obstacles

Proves NP-hardness of minimal execution time planning

Establishes bounds on stretch factor for dense packings

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. Other previous work has largely focused on {\em sequential} schedules, in which one robot moves at a time. We provide constant-factor approximation algorithms for minimizing the execution time of a coordinated, {\em parallel} motion plan for a swarm of robots in the absence of obstacles, provided some amount of separability. Our algorithm achieves {\em constant stretch factor}: If all robots are at most $d$ units from their respective starting positions, the total duration of the overall schedule is $O(d)$. Extensions include unlabeled robots and different classes of robots. We also prove that finding a plan with minimal execution time 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 $\Omega(N^{1/4})$ may be required. On the positive side, we establish a stretch factor of $O(N^{1/2})$ even in this case.