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：我们研究图上一个广义协调运动规划问题的参数化复杂度，其目标是在给定k个机器人集合中，将指定子集的机器人路由至目标位置，以最小化总能量（即总行进长度）。我们提出了新技术，突破了先前局限于实心网格的研究结果。针对仅以k为参数的问题，我们设计了一种固定参数加法近似算法。这一具有独立意义的结果使我们能够证明以下两个关于经典协调运动规划问题的结论：（1）针对单个机器人在避开其他机器人的条件下路由至目标位置的问题（与著名的Rush-Hour Puzzle相关），提出了以k为参数的固定参数算法；（2）针对标准的\textsc{协调运动规划}（CMP）问题，提出了以k加上输入图树宽为参数的固定参数算法，该问题需要将所有k个机器人路由至目标位置。后一结果特别表明，在包括有界高度子网格在内的有界外平面性图上，以k为参数的CMP问题是固定参数可解的。我们通过一个下界结果对上述结论进行补充，该下界排除了以总能量为参数时CMP的固定参数可解性，这与近期在同一参数化下实心网格上该问题的可解性形成对比。作为最终成果，我们将前述固定参数可解性加强到不仅适用于实心网格，而且适用于所有有界局部树宽图——此类图包含但不限于所有有界亏格图。