In this paper, we study a dynamic analogue of the Path Cover problem, which can be solved in polynomial-time in directed acyclic graphs. A temporal digraph has an arc set that changes over discrete time-steps, if the underlying digraph (the union of all the arc sets) is acyclic, then we have a temporal DAG. A temporal path is a directed path in the underlying digraph, such that the time-steps of arcs are strictly increasing along the path. Two temporal paths are temporally disjoint if they do not occupy any vertex at the same time. A temporal (resp. temporally disjoint) path cover is a collection of (resp. temporally disjoint) temporal paths that covers all vertices. In this paper, we study the computational complexities of the problems of finding a temporal (disjoint) path cover with minimum cardinality, denoted as Temporal Path Cover (TPC) and Temporally Disjoint Path Cover (TD-PC). We show that both problems are NP-hard even when the underlying DAG is planar, bipartite, subcubic, and there are only two arc-disjoint time-steps. Moreover, TD-PC remains NP-hard even on temporal oriented trees. In contrast, we show that TPC is polynomial-time solvable on temporal oriented trees by a reduction to Clique Cover for (static undirected) weakly chordal graphs (a subclass of perfect graphs for which Clique Cover admits an efficient algorithm). This highlights an interesting algorithmic difference between the two problems. Although it is NP-hard on temporal oriented trees, TD-PC becomes polynomial-time solvable on temporal oriented lines and temporal rooted directed trees. We also show that TPC (resp. TD-PC) admits an XP (resp. FPT) time algorithm with respect to parameter tmax + tw, where tmax is the maximum time-step, and tw is the treewidth of the underlying static undirected graph.

翻译：本文研究了路径覆盖问题的动态类比，该问题在有向无环图中可在多项式时间内求解。时间有向图的弧集随时间步长离散变化，若其底层有向图（所有弧集的并集）无环，则构成时间有向无环图。时间路径是底层有向图中沿弧的时间步长严格递增的有向路径。两条时间路径若不在同一时间占用任何顶点，则称为时间不相交的。时间路径覆盖（或时间不相交路径覆盖）是一组覆盖所有顶点的（或时间不相交的）时间路径的集合。本文研究了最小基数时间路径覆盖（TPC）和时间不相交路径覆盖（TD-PC）问题的计算复杂性。我们证明，即使底层有向无环图是平面、二分、次立方的，且仅有两个弧不相交的时间步长，这两个问题均为NP难问题。此外，即使在有向时间树上，TD-PC仍保持NP难性。相比之下，我们通过将TPC归约为（静态无向）弱弦图（完美图的一个子类，其团覆盖问题存在高效算法）的团覆盖问题，证明其可在多项式时间内求解有向时间树上的TPC。这凸显了这两个问题间有趣的算法差异。尽管TD-PC在有向时间树上NP难，但在有向时间线及有向时间根树上可多项式时间求解。我们还表明，TPC（或TD-PC）在参数 tmax + tw 下存在XP（或FPT）时间算法，其中 tmax 为最大时间步长，tw 为底层静态无向图的树宽。