柯尼斯堡观光：时间图中的欧拉行走
Königsberg Sightseeing: Eulerian Walks in Temporal Graphs
摘要：
Eulerian walk（或Eulerian trail）是一种行走（resp）。路径），其至少访问图$G$的每条边（分别确切地）一次。这个概念首先由莱昂哈德·欧拉在1736年解决著名的柯尼斯堡七桥问题时讨论。如果欧拉不得不坐公共汽车呢？在具有$λ: E(G)\to 2^{[τ]}$的时间图$(G,λ)$中，边$e\in E(G)$仅在由$λ(e)\subseteq [τ]$指定的时间可用，同样地，城市的公共交通网络或观光旅游的连接仅在预定时间可用。在这种情况下，尽管欧拉路径和行走的几种翻译在时间上是可能的，但文献中只利用了一种非常特殊的变化，特别是对于无限动态网络（Orlin，1984）。在本文中，我们处理的时间行走，本地的线索，和线索，分别指的是没有约束的边遍历，约束到不重复相同的边缘在一个单一的时间戳，并约束到永远不会重复相同的边缘在整个遍历。我们表明，如果边缘总是可用的，然后决定是否$(G,λ)$有一个时间的步行或线索是多项式，而决定是否有一个本地的线索是NP完全的，即使它有生命周期~2。相反，在一般情况下，解决这些问题中的任何一个都是NP完全的，即使在非常严格的假设下。
英文摘要：
An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph $G$ at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. What if Euler had to take a bus? In a temporal graph $(G,λ)$, with $λ: E(G)\to 2^{[τ]}$, an edge $e\in E(G)$ is available only at the times specified by $λ(e)\subseteq [τ]$, in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this scenario, even though several translations of Eulerian trails and walks are possible in temporal terms, only a very particular variation has been exploited in the literature, specifically for infinite dynamic networks (Orlin, 1984). In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether $(G,λ)$ has a temporal walk or trail is polynomial, while deciding whether it has a local trail is NP-complete even if it has lifetime~2. In contrast, in the general case, solving any of these problems is NP-complete, even under very strict hypothesis.
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