Time-evolving or temporal graphs gain more and more popularity when studying the behavior of complex networks. In this context, the multistage view on computational problems is among the most natural frameworks. Roughly speaking, herein one studies the different (time) layers of a temporal graph (effectively meaning that the edge set may change over time, but the vertex set remains unchanged), and one searches for a solution of a given graph problem for each layer. The twist in the multistage setting is that the solutions found must not differ too much between subsequent layers. We relax on this already established notion by introducing a global instead of the local budget view studied so far. More specifically, we allow for few disruptive changes between subsequent layers but request that overall, that is, summing over all layers, the degree of change is moderate. Studying several classical graph problems (both NP-hard and polynomial-time solvable ones) from a parameterized complexity angle, we encounter both fixed-parameter tractability and parameterized hardness results. Somewhat surprisingly, we find that sometimes the global multistage versions of NP-hard problems such as Vertex Cover turn out to be computationally more tractable than the ones of polynomial-time solvable problems such as Matching.

中文翻译：

---

全球预算的多阶段图问题

在研究复杂网络的行为时，时间演化图或时间图越来越受欢迎。在这种情况下，对计算问题的多阶段观点是最自然的框架之一。粗略地说，这里研究时间图的不同（时间）层（实际上意味着边集可能随时间变化，但顶点集保持不变），并为每一层寻找给定图问题的解决方案. 多级设置中的扭曲是找到的解决方案在后续层之间不能有太大差异。我们通过引入全球而不是迄今为止研究的本地预算视图来放松这个已经建立的概念。更具体地说，我们允许在后续层之间进行很少的破坏性更改，但要求总体上，即对所有层求和，变化程度适中。从参数化复杂性的角度研究几个经典的图问题（NP 难和多项式时间可解的问题），我们遇到了固定参数易处理性和参数化硬度结果。有点令人惊讶的是，我们发现有时像​​ Vertex Cover 这样的 NP-hard 问题的全局多阶段版本在计算上比多项式时间可解决的问题（如匹配）更容易处理。