On the complexity of constrained reconfiguration and motion planning

TL;DR

This paper investigates the computational complexity of reconfiguration and motion planning problems involving multiple agents with constrained actions, proving NP-completeness in general but providing efficient algorithms for special cases.

Contribution

It establishes NP-completeness of the k-Compatible Ordering problem and offers polynomial algorithms for specific constrained scenarios, extending the understanding of motion planning complexity.

Findings

k-Compatible Ordering is NP-complete even in planar or acyclic cases

Polynomial algorithms exist for k=1 and bounded treewidth graphs

Generalized variants support multiple actions per agent

Abstract

Coordinating the motion of multiple agents in constrained environments is a fundamental challenge in robotics, motion planning, and scheduling. A motivating example involves $n$ robotic arms, each represented as a line segment. The objective is to rotate each arm to its vertical orientation, one at a time (clockwise or counterclockwise), without collisions nor rotating any arm more than once. This scenario is an example of the more general $k$-Compatible Ordering problem, where $n$ agents, each capable of $k$ state-changing actions, must transition to specific target states under constraints encoded as a set $\mathcal{G}$ of $k$ pairs of directed graphs. We show that $k$-Compatible Ordering is $\mathsf{NP}$-complete, even when $\mathcal{G}$ is planar, degenerate, or acyclic. On the positive side, we provide polynomial-time algorithms for cases such as when $k = 1$ or $\mathcal{G}$ has bounded treewidth. We also introduce generalized variants supporting multiple state-changing actions per agent, broadening the applicability of our framework. These results extend to a wide range of scheduling, reconfiguration, and motion planning applications in constrained environments.