The pebble motion on trees (PMT) problem consists in finding a feasible sequence of moves that repositions a set of pebbles to assigned target vertices. This problem has been widely studied because, in many cases, the more general Multi-Agent path finding (MAPF) problem on graphs can be reduced to PMT. We propose a simple and easy to implement procedure, which finds solutions of length O(knc + n^2), where n is the number of nodes, $k$ is the number of pebbles, and c the maximum length of corridors in the tree. This complexity result is more detailed than the current best known result O(n^3), which is equal to our result in the worst case, but does not capture the dependency on c and k.

翻译：树上卵石移动（PMT）问题旨在寻找一系列可行移动序列，将一组卵石重新定位至指定的目标顶点。该问题已被广泛研究，因为许多情况下，图上更一般的多智能体路径规划（MAPF）问题可简化为PMT。我们提出一种简单且易于实现的算法，该算法能够求解长度为O(knc + n²)的路径，其中n为节点数，k为卵石数，c为树中走廊的最大长度。这一复杂度结果比当前已知的最佳结果O(n³)更为精细——在最坏情况下，我们的结果与O(n³)相当，但无法捕捉对c和k的依赖关系。