The Parameterized Complexity of Coordinated Motion Planning

Eduard Eiben, Robert Ganian, Iyad Kanj

Research output: Chapter in Book/Report/Conference proceedingConference contribution


In Coordinated Motion Planning (CMP), we are given a rectangular-grid on which k robots occupy k distinct starting gridpoints and need to reach k distinct destination gridpoints. In each time step, any robot may move to a neighboring gridpoint or stay in its current gridpoint, provided that it does not collide with other robots. The goal is to compute a schedule for moving the k robots to their destinations which minimizes a certain objective target---prominently the number of time steps in the schedule, i.e., the makespan, or the total length traveled by the robots. We refer to the problem arising from minimizing the former objective target as CMP-M and the latter as CMP-L. Both CMP-M and CMP-L are fundamental problems that were posed as the computational geometry challenge of SoCG 2021, and CMP also embodies the famous
(n^2-1)-puzzle as a special case.

In this paper, we settle the parameterized complexity of CMP-M and CMP-L with respect to their two most fundamental parameters: the number of robots, and the objective target. We develop a new approach to establish the fixed-parameter tractability of both problems under the former parameterization that relies on novel structural insights into optimal solutions to the problem. When parameterized by the objective target, we show that CMP-L remains fixed-parameter tractable while CMP-M becomes para-NP-hard. The latter result is noteworthy, not only because it improves the previously-known boundaries of intractability for the problem, but also because the underlying reduction allows us to establish---as a simpler case---the NP-hardness of the classical Vertex Disjoint and Edge Disjoint Paths problems with constant path-lengths on grids.
Original languageEnglish
Title of host publicationThe 39th International Symposium on Computational Geometry (SoCG 2023)
PublisherSchloss Dagstuhl--Leibniz-Zentrum fuer Informatik
Number of pages16
ISBN (Electronic)978-3-95977-273-0
Publication statusPublished - 9 Jun 2023
EventThe 39th International Symposium on Computational Geometry - University of Texas at Dallas, Dallas, United States
Duration: 12 Jun 202315 Jun 2023


ConferenceThe 39th International Symposium on Computational Geometry
Abbreviated titleSoCG 2023
Country/TerritoryUnited States
Internet address

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