Time-evolving or temporal graphs gain more and more popularity when exploring 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. 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. In addition to time complexity, we also analyze the space efficiency of our algorithms.

当探索复杂的网络时，时间演化图或时态图越来越受欢迎。在这种情况下，关于计算问题的多阶段观点是最自然的框架之一。粗略地说，本文研究的是时间图的不同（时间）层（有效地意味着边集可能随时间而变化，但顶点集保持不变），然后人们针对每一层寻找给定图问题的解决方案。多阶段设置中的问题是，在后续各层之间找到的解决方案之间的差异不得太大。我们通过引入全球而不是到目前为止研究的地方预算观点。更具体地说，我们允许在随后的各层之间进行很少的破坏性更改，但要求总体（即，对所有层进行汇总）更改程度是适度的。从参数化的复杂性角度研究几个经典的图形问题（NP-hard和多项式时间可解问题），我们同时遇到了固定参数易处理性和参数化硬度结果。令人惊讶的是，我们发现，有时诸如难点顶点覆盖之类的NP难问题的全局多级版本比诸如Matching等多项式时间可解决的问题在计算上更易于处理。除了时间复杂度，我们还分析了算法的空间效率。