Coordinated Motion Planning is FPT on Discretized Simple Polygons

2026-05-08

AI Summary

This study addresses the problem of collision-free cooperative path planning for multiple robots operating on graphs derived from the discretization of simple polygons, with the objective of minimizing the total travel distance across all robots. The work proposes a fixed-parameter tractable (FPT) algorithm parameterized by the number of robots \(k\), thereby establishing—for the first time—that this problem is FPT on such polygon-induced graphs. By leveraging structural properties of the underlying graph and geometric constraints inherent to simple polygons, the approach extends existing FPT results from grid environments to more general planar, obstacle-constrained settings. This advancement provides significant progress toward resolving open questions concerning subgrid and planar graph variants of multi-robot path planning.

Abstract

In the coordinated motion planning problem, we are given a graph together with the starting and destination vertices of $k$ robots. At each time step, any subset of robots may move, each traversing an edge of the graph, provided that no two robots collide. The goal is to compute a schedule that routes all robots to their destinations while minimizing some objective function. In this paper, we focus on the well-studied objective of minimizing the total travel length of all robots. This problem is known to be NP-hard, and it has been shown to be fixed-parameter tractable (FPT), when parameterized by the number $k$ of robots, on full grids (SoCG 2023) and on bounded-treewidth graphs (ICALP 2024). We present a fixed-parameter algorithm for coordinated motion planning, parameterized by the number $k$ of robots, on graphs arising from discretizations of simple polygons. Such graphs are of particular interest in real-world applications, where planar motion is often constrained to discretized representations of polygonal environments. Moreover, these graphs generalize rectangular grids; consequently, our result constitutes a significant step toward resolving the parameterized complexity of coordinated motion planning on subgrids and, ultimately, planar graphs -- two prominent open problems in the field.

Problem

• Coordinated Motion Planning
• Simple Polygons
• Fixed-Parameter Tractability
• Robot Navigation
• Collision Avoidance

Innovation

• coordinated motion planning
• fixed-parameter tractability
• discretized simple polygons
• robot navigation
• parameterized complexity