In this work we extend the concept of out/in-branchings spanning the vertices of a digraph (also called directed spanning trees) to temporal graphs, which are digraphs where arcs are available only at prescribed times. While the literature has focused on minimum weight/earliest arrival time Temporal Out-Branchings (TOB), we solve the problem for other optimization criteria. In particular, we define five different types of TOBs based on the optimization of the travel duration (FT-TOB), of the departure time (LD-TOB), of the number of transfers (MT-TOB), of the total waiting time (MW-TOB), and of the travelling time (ST-TOB). For D$\in \{$LD,MT,ST$\}$, we provide necessary and sufficient conditions for the existence of a spanning D-TOB; when it does not exist, we characterize the maximum vertex set that a D-TOB can span. Moreover, we provide a log linear algorithm for computing such branchings. For D$\in \{$FT,MW$\}$, we prove that deciding the existence of a spanning D-TOB is NP-complete; we also show that the same results hold for optimal temporal in-branchings. Finally, we investigate the related problem of computing a spanning temporal subgraph with the minimum number of arcs and optimizing a chosen criterion D. This problem turns out to be NP-hard for any D. The hardness results are quite surprising, as computing optimal paths between nodes can always be done in polynomial time.

翻译：本文我们将出/入分支（覆盖有向图所有顶点的有向生成树）的概念推广到时态图（即弧仅在指定时间可用的有向图）。现有文献主要关注最小权重/最早到达时间的时态外出分支（TOB），而我们则针对其他优化准则求解该问题。具体地，我们基于旅行时长优化、出发时间优化、换乘次数优化、总等待时间优化和行驶时间优化定义了五种不同类型的TOB：FT-TOB、LD-TOB、MT-TOB、MW-TOB和ST-TOB。对于D∈{LD,MT,ST}，我们给出了生成D-TOB存在的充要条件；当不存在时，我们刻画了D-TOB所能覆盖的最大顶点集。此外，我们提出了一种对数线性算法来计算此类分支。对于D∈{FT,MW}，我们证明判定生成D-TOB的存在性是NP完全的；同时表明相同结论对最优时态入分支也成立。最后，我们研究了相关的最小弧数生成时态子图问题，并对选定准则D进行优化。该问题对任意D均为NP难的。这些困难结果令人惊讶，因为节点间最优路径的计算总能在多项式时间内完成。