Coordinated Motion Planning is FPT on Discretized Simple Polygons

TL;DR

This paper presents a fixed-parameter algorithm for coordinated motion planning on graphs derived from discretized simple polygons, advancing the understanding of the problem's complexity in planar environments.

Contribution

The authors develop an FPT algorithm for motion planning on discretized simple polygons, extending previous results on grids and bounded-treewidth graphs.

Findings

The algorithm is fixed-parameter tractable with respect to the number of robots.

It applies to graphs from discretized simple polygons, common in real-world planar environments.

This advances the understanding of motion planning complexity on planar graphs.

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.