In a Subgraph Problem we are given some graph and want to find a feasible subgraph that optimizes some measure. We consider Multistage Subgraph Problems (MSPs), where we are given a sequence of graph instances (stages) and are asked to find a sequence of subgraphs, one for each stage, such that each is optimal for its respective stage and the subgraphs for subsequent stages are as similar as possible. We present a framework that provides a $(1/\sqrt{2\chi})$-approximation algorithm for the $2$-stage restriction of an MSP if the similarity of subsequent solutions is measured as the intersection cardinality and said MSP is preficient, i.e., we can efficiently find a single-stage solution that prefers some given subset. The approximation factor is dependent on the instance's intertwinement $\chi$, a similarity measure for multistage graphs. We also show that for any MSP, independent of similarity measure and preficiency, given an exact or approximation algorithm for a constant number of stages, we can approximate the MSP for an unrestricted number of stages. Finally, we combine and apply these results and show that the above restrictions describe a very rich class of MSPs and that proving membership for this class is mostly straightforward. As examples, we explicitly state these proofs for natural multistage versions of Perfect Matching, Shortest s-t-Path, Minimum s-t-Cut and further classical problems on bipartite or planar graphs, namely Maximum Cut, Vertex Cover, Independent Set, and Biclique.

翻译：在子图问题中，给定一个图，我们希望找到一个可行子图以优化某个度量。我们考虑多阶段子图问题（MSPs），其中给定一系列图实例（阶段），要求找到一系列子图（每个阶段对应一个子图），使得每个子图在其对应阶段中是最优的，并且后续阶段的子图尽可能相似。我们提出一个框架，该框架为MSP的2阶段限制提供了一个$(1/\sqrt{2\chi})$近似算法，前提是后续解的相似性通过交集基数衡量，且该MSP是“优选的”（即我们能高效找到优先考虑某给定子集的单阶段解）。近似因子取决于实例的“交织度”$\chi$，这是多阶段图的一种相似性度量。我们还证明，对于任何MSP（无论相似性度量或“优选性”如何），若给定常数个阶段上的精确或近似算法，我们就能近似无限制阶段数的MSP。最后，我们结合并应用这些结果，证明上述限制描述了一个非常丰富的MSP类，且验证该类成员资格通常是直接的。作为示例，我们明确陈述了针对完美匹配、最短s-t路径、最小s-t割以及二分图或平面图上的经典问题（即最大割、顶点覆盖、独立集和双团）的自然多阶段版本的这些证明。