Connected Coordinated Motion Planning with Bounded Stretch

TL;DR

This paper studies the complex problem of coordinated motion planning for large robot groups, introducing algorithms with constant-factor approximations for bounded scale and stretch, and analyzing the problem's computational hardness.

Contribution

It establishes NP-completeness results, provides polynomial-time checks for specific makespan cases, and develops algorithms with constant stretch and scale factors for coordinated robot motion.

Findings

NP-complete for makespan 2, polynomial for makespan 1

Algorithms achieve constant stretch and scale factors

Schedule duration is proportional to the maximum Manhattan distance d

Abstract

We consider the problem of connected coordinated motion planning for a large collective of simple, identical robots: From a given start grid configuration of robots, we need to reach a desired target configuration via a sequence of parallel, collision-free robot motions, such that the set of robots induces a connected grid graph at all integer times. The objective is to minimize the makespan of the motion schedule, i.e., to reach the new configuration in a minimum amount of time. We show that this problem is NP-complete, even for deciding 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. On the algorithmic side, we establish simultaneous constant-factor approximation for two fundamental parameters, by achieving constant stretch for constant scale. Scaled shapes (which arise by increasing all dimensions of a given object by the same multiplicative factor) have been considered in previous seminal work on self-assembly, often with unbounded or logarithmic scale factors; we provide methods for a generalized scale factor, bounded by a constant. Moreover, our algorithm achieves a constant stretch factor: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of $d$, then the total duration of our overall schedule is $\mathcal{O}(d)$, which is optimal up to constant factors.