Algorithms and Complexity for Path Covers of Temporal DAGs: when is Dilworth Dynamic?

In this paper, we study a dynamic analogue of the Path Cover problem, whichcan be solved in polynomial-time in directed acyclic graphs. A temporal digraphhas 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. Atemporal path is a directed path in the underlying digraph, such that thetime-steps of arcs are strictly increasing along the path. Two temporal pathsare temporally disjoint if they do not occupy any vertex at the same time. Atemporal (resp. temporally disjoint) path cover is a collection of (resp.temporally disjoint) temporal paths that covers all vertices. In this paper, westudy 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 problemsare NP-hard even when the underlying DAG is planar, bipartite, subcubic, andthere are only two arc-disjoint time-steps. Moreover, TD-PC remains NP-hardeven on temporal oriented trees. In contrast, we show that TPC ispolynomial-time solvable on temporal oriented trees by a reduction to CliqueCover for (static undirected) weakly chordal graphs (a subclass of perfectgraphs for which Clique Cover admits an efficient algorithm). This highlightsan interesting algorithmic difference between the two problems. Although it isNP-hard on temporal oriented trees, TD-PC becomes polynomial-time solvable ontemporal oriented lines and temporal rooted directed trees. We also show thatTPC (resp. TD-PC) admits an XP (resp. FPT) time algorithm with respect toparameter tmax + tw, where tmax is the maximum time-step, and tw is thetreewidth of the underlying static undirected graph.