In a temporal network with discrete time-labels on its edges, entities and information can only "flow" along sequences of edges whose time-labels are non-decreasing (resp. increasing), i.e. along temporal (resp. strict temporal) paths. Nevertheless, in the model for temporal networks of [Kempe et al., JCSS, 2002], the individual time-labeled edges remain undirected: an edge $e=\{u,v\}$ with time-label $t$ specifies that "$u$ communicates with $v$ at time $t$". This is a symmetric relation between $u$ and $v$, and it can be interpreted that the information can flow in either direction. In this paper we make a first attempt to understand how the direction of information flow on one edge can impact the direction of information flow on other edges. More specifically, we introduce the notion of a temporal transitive orientation and we systematically investigate its algorithmic behavior in various situations. An orientation of a temporal graph is called temporally transitive if, whenever $u$ has a directed edge towards $v$ with time-label $t_1$ and $v$ has a directed edge towards $w$ with time-label $t_2\geq t_1$, then $u$ also has a directed edge towards $w$ with some time-label $t_3\geq t_2$. If we just demand that this implication holds whenever $t_2 > t_1$, the orientation is called strictly temporally transitive. Our main result is a conceptually simple, yet technically quite involved, polynomial-time algorithm for recognizing whether a given temporal graph $\mathcal{G}$ is transitively orientable. In wide contrast we prove that, surprisingly, it is NP-hard to recognize whether $\mathcal{G}$ is strictly transitively orientable. Additionally we introduce and investigate further related problems to temporal transitivity, notably among them the temporal transitive completion problem, for which we prove both algorithmic and hardness results.

中文翻译：

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传递定向时间图的复杂性

在其边缘具有离散时间标签的时间网络中，实体和信息只能沿其时间标签不减少（分别增加）的边缘序列“流动”，即沿着时间（严格时间限制）路径“流动”。然而，在[Kempe等人，JCSS，2002]的时态网络模型中，各个带有时间标签的边仍然是无方向的：带有时间标签$ t $的边$ e = \ {u，v \} $ “ $ u $在时间$ t $与$ v $通信”。这是$ u $和$ v $之间的对称关系，可以解释为信息可以沿任一方向流动。在本文中，我们首先尝试了解一个边缘上的信息流的方向如何影响另一边缘上的信息流的方向。进一步来说，我们介绍了时间可及取向的概念，并系统地研究了它在各种情况下的算法行为。如果每当$ u $具有指向带有时间标签$ t_1 $的$ v $的有向边且$ v $具有带有带有时间标签$ t_2 \的$ w $的有向边时，时间图的方向称为时间传递。 geq t_1 $，然后$ u $也具有指向$ w $的定向边缘，并带有一些时间标签$ t_3 \ geq t_2 $。如果仅当$ t_2> t_1 $时要求这种含义成立，则该方向严格称为时间传递。我们的主要结果是一个概念上简单但在技术上相当复杂的多项式时间算法，用于识别给定的时间图$ \ mathcal {G} $是否可传递。与之形成鲜明对比的是，我们证明，NP很难识别$ \ mathcal {G} $是否严格可传递。此外，我们介绍并研究了与时间传递性相关的其他问题，尤其是时间传递性完成问题，为此我们证明了算法和硬度结果。