We study the network untangling problem introduced by Rozenshtein et al. (Data Min. Knowl. Disc. 35(1), 213–247, 2021), which is a variant of Vertex Cover on temporal graphs–graphs whose edge set changes over discrete time steps. They introduce two problem variants. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks. Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability.

我们研究了 Rozenshtein 等人提出的网络理清问题。（Data Min. Knowl. Disc. 35(1), 213–247, 2021），这是时间图 上顶点覆盖的变体，时间图的边缘集随离散时间步长变化。他们引入了两种问题变体。目标是为每个顶点选择最多k 个时间间隔，以便覆盖所有时间边，并且（取决于问题变体）最大间隔长度或间隔长度的总和最小化。这个问题具有数据挖掘应用程序来查找解释复杂网络中实体交互的活动时间线。该问题的两种变体都是 NP 困难的。在本文中，我们发起了涉及以下参数的多元复杂性分析：顶点数量、时间图的生命周期、每个顶点的间隔数量以及间隔长度界限。对于这两个问题版本，我们（几乎）完全解决了这四个参数的所有组合的参数化复杂性，从而描绘了固定参数易处理性的边界。