Several classical combinatorial problems have been considered and analysed on temporal graphs. Recently, a variant of Vertex Cover on temporal graphs, called MinTimelineCover, has been introduced to summarize timeline activities in social networks. The problem asks to cover every temporal edge while minimizing the total span of the vertices (where the span of a vertex is the length of the timestamp interval it must remain active in, minus one). While the problem has been shown to be NP-hard even in very restricted cases, its parameterized complexity has not been fully understood. The problem is known to be in FPT under the span parameter only for graphs with two timestamps, but the parameterized complexity for the general case is open. We settle this open problem by giving an FPT algorithm that is based on a combination of iterative compression and a reduction to the Digraph Pair Cut problem, a powerful problem that has received significant attention recently.

翻译：多个经典组合问题已在时序图上得到研究与分析。近期，一类名为最小时间线覆盖（MinTimelineCover）的时序图顶点覆盖变体被提出，用于归纳社交网络中的时间线活动。该问题要求覆盖所有时序边，同时最小化顶点的总跨度（其中，顶点的跨度指其必须保持活跃的时间戳区间长度减一）。尽管该问题在极其受限的情况下已被证明为NP难问题，其参数化复杂性尚未完全明确。已知仅在具有两个时间戳的图中，该问题在跨度参数下属于FPT类，但一般情形下的参数化复杂性仍是开放问题。我们通过提出一种基于迭代压缩与有向图对切（Digraph Pair Cut）问题归约相结合的FPT算法，解决了这一开放问题——后者是一种近期备受关注且功能强大的问题。