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.

翻译：我们研究图上一类广义协调运动规划问题的参数化复杂度，目标是在给定$k$个机器人中路由指定子集到其目的地，以最小化总能量（即总运动距离）。我们开发了新颖技术，突破了此前局限于实心网格的结果。针对仅以$k$为参数化的问题，我们设计了一种固定参数加性近似算法。这一具有独立意义的结果使我们得以证明关于经典协调运动规划问题的以下两个结论：（1）以$k$为参数化的固定参数算法，用于在避开其他机器人的情况下将单个机器人路由至目的地，这一结果与著名的“高峰时段谜题”相关；（2）以$k$与输入图树宽为参数化的固定参数算法，用于标准\textsc{协调运动规划}（CMP）问题——需将所有$k$个机器人路由至其目的地。后者结果尤其表明，在包含有界高度子网格的有界外平面性图上，CMP关于$k$具有固定参数可解性。我们进一步以排除性下界补充上述结果，该下界否定了CMP关于总能量参数化的固定参数可解性。这与近期在实心网格上同类参数化下获得的可解性结果形成对比。作为最终结论，我们将前述固定参数可解性不仅推广至实心网格，更覆盖所有有界局部树宽图——该类图包含如所有有界亏格图等。