Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy
<|reference_start|>Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy: We study the parameterized complexity of a generalization of the coordinated motion planning problem on graphs, where the goal is to route a specified subset of a given set of $k$ robots to their destinations with the aim of minimizing the total energy (i.e., the total length traveled). We develop novel techniques to push beyond previously-established results that were restricted to solid grids.   We design a fixed-parameter additive approximation algorithm for this problem parameterized by $k$ alone. This result, which is of independent interest, allows us to prove the following two results pertaining to well-studied coordinated motion planning problems: (1) A fixed-parameter algorithm, parameterized by $k$, for routing a single robot to its destination while avoiding the other robots, which is related to the famous Rush-Hour Puzzle; and (2) a fixed-parameter algorithm, parameterized by $k$ plus the treewidth of the input graph, for the standard \textsc{Coordinated Motion Planning} (CMP) problem in which we need to route all the $k$ robots to their destinations. The latter of these results implies, among others, the fixed-parameter tractability of CMP parameterized by $k$ on graphs of bounded outerplanarity, which include bounded-height subgrids.   We complement the above results with a lower bound which rules out the fixed-parameter tractability for CMP when parameterized by the total energy. This contrasts the recently-obtained tractability of the problem on solid grids under the same parameterization. As our final result, we strengthen the aforementioned fixed-parameter tractability to hold not only on solid grids but all graphs of bounded local treewidth -- a class including, among others, all graphs of bounded genus.<|reference_end|>
@article{deligkas2024parameterized,
    title={Parameterized Algorithms for Coordinated Motion Planning: Minimizing
  Energy},
    author={Argyrios Deligkas and Eduard Eiben and Robert Ganian and Iyad Kanj and
  M. S. Ramanujan},
    journal={arXiv preprint arXiv:2404.15950},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.15950},
    primaryClass={cs.DM}
}