On the complexity of constrained reconfiguration and motion planning

2025-08-18

AI Summary

This paper investigates the coordinated reconfiguration of multi-agent systems under graph-structured constraints, exemplified by the problem of rotating n segment-like robotic arms sequentially to a vertical orientation without collisions. We propose a unified modeling framework—k-Compatible Ordering—in which each agent undergoes exactly k state transitions, subject to temporal and conflict constraints encoded by k directed graphs. We prove that the general problem is NP-complete. For special graph classes—including planar graphs and graphs of bounded treewidth—we design polynomial-time decision and constructive algorithms. Our approach integrates computational complexity analysis, graph-theoretic modeling, and dynamic programming, employing a state-transition graph to characterize motion feasibility. The contributions include: (i) establishing tight computational complexity bounds for the problem; and (ii) providing scalable theoretical tools and efficient algorithms for robot reconfiguration, task scheduling, and path planning in constrained environments.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Intelligent Robots: Motion and Path Planning
• Multiagent Systems: Multiagent Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Coordinating multi-agent motion in constrained environments
• Solving NP-complete k-Compatible Ordering problem efficiently
• Developing algorithms for constrained scheduling and reconfiguration

Innovation

• NP-complete analysis for k-Compatible Ordering
• Polynomial-time algorithms for bounded treewidth
• Generalized variants for multiple agent actions