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：我们研究了图上一个协调运动规划问题的推广形式的参数化复杂度，该问题的目标是在最小化总能量（即总行进长度）的前提下，将给定k个机器人中的指定子集路由至其目的地。我们开发了新技术，突破了先前局限于实心网格的研究结果。针对该问题，我们设计了一个仅以k为参数的固定参数加性近似算法。这一具有独立意义的结果使我们能够证明以下两个关于被广泛研究的协调运动规划问题的结论：(1) 一个以k为参数的固定参数算法，用于在避开其他机器人的情况下将单个机器人路由至其目的地，该问题与著名的"塞车时间"（Rush-Hour）益智游戏相关；(2) 一个以k加上输入图的树宽为参数的固定参数算法，用于标准的\textsc{协调运动规划}（CMP）问题，即需要将所有k个机器人路由至各自目的地。后一结果尤其意味着，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问题是固定参数可处理的。我们通过一个下界结果对上述结论进行了补充，该下界排除了以总能量为参数时CMP的固定参数可处理性。这与最近在同一参数化下实心网格上该问题的可处理性形成了对比。作为最终结果，我们将前述固定参数可处理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——此类图包括（但不限于）所有有界亏格图。