时间图传递定向的复杂性
The Complexity of Transitively Orienting Temporal Graphs
AI总结：
本文引入了时间图的时间传递定向概念，证明识别时间可传递定向存在多项式时间算法，而识别严格时间可传递定向是NP-hard的，并研究了相关的时间传递完成问题。
摘要：
在边具有离散时间标签的时间网络中，实体和信息只能沿着时间标签非递减（相应地，递增）的边序列“流动”，即沿着时间（相应地，严格时间）路径流动。然而，在 [Kempe, Kleinberg, Kumar, JCSS, 2002] 的时间网络模型中，单个带时间标签的边仍然是无向的：一条带有时间标签 $t$ 的边 $e=\{u,v\}$ 规定“$u$ 在时间 $t$ 与 $v$ 通信”。在本文中，我们首次尝试理解一条边上的信息流方向如何影响其他边上的信息流方向。更具体地说，自然地扩展静态图中传递定向的经典概念，我们引入了时间传递定向的基本概念，并系统地研究了其算法行为。时间图的一个定向被称为时间传递的，如果每当 $u$ 有一条带有时间标签 $t_1$ 的有向边指向 $v$，且 $v$ 有一条带有时间标签 $t_2\geq t_1$ 的有向边指向 $w$ 时，$u$ 也有一条带有某个时间标签 $t_3\geq t_2$ 的有向边指向 $w$。如果我们只要求该蕴含关系在 $t_2 > t_1$ 时成立，我们则称该定向为严格时间传递的，因为它是基于从 $u$ 到 $w$ 的严格有向时间路径的。我们的主要结果是一个概念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 $\mathcal{G}$ 是否是可传递定向的。形成鲜明对比的是，我们令人惊讶地证明，识别 $\mathcal{G}$ 是否是严格可传递定向的是 NP-hard 的。此外，我们引入并研究了与时间传递性进一步相关的问题，其中值得注意的是时间传递完成问题，对此我们证明了算法和难解性结果。
英文摘要：
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, Kleinberg, Kumar, 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 $t$''. 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, naturally extending the classical notion of a transitive orientation in static graphs, we introduce the fundamental notion of a temporal transitive orientation and we systematically investigate its algorithmic behavior. 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 $t_2 > t_1$, we call the orientation strictly temporally transitive, as it is based on the strict directed temporal path from $u$ to $w$. 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.
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