A temporal graph is a graph in which edges are assigned a time label. Two nodes u and v of a temporal graph are connected one to the other if there exists a path from u to v with increasing edge time labels. We consider the problem of assigning time labels to the edges of a digraph in order to maximize the total reachability of the resulting temporal graph (that is, the number of pairs of nodes which are connected one to the other). In particular, we prove that this problem is NP-hard. We then conjecture that the problem is approximable within a constant approximation ratio. This conjecture is a consequence of the following graph theoretic conjecture: any strongly connected directed graph with n nodes admits an out-arborescence and an in-arborescence that are edge-disjoint, have the same root, and each spans $\Omega$(n) nodes.

翻译：时间图是一种边被赋予时间标签的图。时间图中的两个节点u和v若存在一条从u到v且边时间标签递增的路径，则称两者相互连通。我们考虑为有向图的边分配时间标签，以最大化生成时间图的总可达性（即相互连通的节点对数）。特别地，我们证明该问题是NP难的。进一步我们猜想该问题存在常数近似比的可近似性，这一猜想源于以下图论猜想：任意包含n个节点的强连通有向图均存在边不交、根相同且各自覆盖Ω(n)个节点的外出分支树与内入分支树。