In Graph Coordinated Motion Planning, we are given a graph G some of whose vertices are occupied by robots, and we are asked to route k marked robots to their destinations while avoiding collisions and without exceeding a given budget 𝓁 on the number of robot moves. We continue the recent investigation of the problem [ICALP 2024], focusing on the parameter k that captures the task of routing a small number of robots in a possibly crowded graph. We prove that the problem is W[1]-hard parameterized by 𝓁 even for k = 1, but fixed-parameter tractable parameterized by k plus the treedepth of G. We complement the latter algorithm with an NP-hardness reduction which shows that both parameters are necessary to achieve tractability.
@InProceedings{deligkas_et_al:LIPIcs.WADS.2025.20, author = {Deligkas, Argyrios and Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Leko, Dominik and Ramanujan, M. S.}, title = {{Routing Few Robots in a Crowded Network}}, booktitle = {19th International Symposium on Algorithms and Data Structures (WADS 2025)}, pages = {20:1--20:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-398-0}, ISSN = {1868-8969}, year = {2025}, volume = {349}, editor = {Morin, Pat and Oh, Eunjin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.20}, URN = {urn:nbn:de:0030-drops-242516}, doi = {10.4230/LIPIcs.WADS.2025.20}, annote = {Keywords: graph coordinated motion planning, parameterized complexity, treedepth} }
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