We study the algorithmic complexity of partitioning the vertex set of a given (di)graph into a small number of paths. The Path Partition problem (PP) has been studied extensively, as it includes Hamiltonian Path as a special case. The natural variants where the paths are required to be either \emph{induced} (Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition, SPP), have received much less attention. Both problems are known to be NP-complete on undirected graphs; we strengthen this by showing that they remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP remains \NP-hard on undirected bipartite graphs. When parameterized by the natural parameter ``number of paths'', both SPP and IPP are shown to be W{1}-hard on DAGs. We also show that SPP is in \XP both for DAGs and undirected graphs for the same parameter, as well as for other special subclasses of directed graphs (IPP is known to be NP-hard on undirected graphs, even for two paths). On the positive side, we show that for undirected graphs, both problems are in FPT, parameterized by neighborhood diversity. We also give an explicit algorithm for the vertex cover parameterization of PP. When considering the dual parameterization (graph order minus number of paths), all three variants, IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the mentioned neighborhood diversity and dual parameterization results to directed graphs; here, we need to define a proper novel notion of directed neighborhood diversity. As we also show, most of our results also transfer to the case of covering by edge-disjoint paths, and purely covering.

翻译：我们研究了将给定（有向）图的顶点集划分为少量路径的算法复杂性。路径划分问题（PP）已被广泛研究，因为它将哈密顿路径作为特例包含在内。自然变体要求路径要么是**导出**（导出路径划分，IPP）要么是**最短**（最短路径划分，SPP），但这两者受到的关注较少。已知这两个问题在无向图上均为NP完全问题；我们通过证明它们在平面二分有向无环图（DAG）上仍保持NP完全性，以及SPP在无向二分图上仍为NP难问题，强化了这一结论。当以自然参数“路径数量”进行参数化时，SPP和IPP在DAG上均被证明是W[1]难的。我们还表明，对于相同参数，SPP在DAG和无向图上均属于XP类，并且对其他特殊有向图子类也成立（已知IPP在无向图上即使对于两条路径也是NP难的）。积极方面，我们证明对于无向图，这两个问题在邻域多样性参数化下均属于FPT类。我们还为PP的顶点覆盖参数化给出了显式算法。当考虑对偶参数化（图阶减去路径数量）时，IPP、SPP和PP这三个变体在无向图上均被证明属于FPT类。我们还将上述邻域多样性和对偶参数化结果推广到有向图；为此，我们需要定义一种新颖的适当的有向邻域多样性概念。我们还证明，我们的大部分结果也适用于边不相交路径覆盖和纯覆盖情形。