A general approximation for multistage subgraph problems

Subgraph Problems are optimization problems on graphs where a solution is a subgraph that satisfies some property and optimizes some measure. Examples include shortest path, minimum cut, maximum matching, or vertex cover. In reality, however, one often deals with time-dependent data, i.e., the input graph may change over time and we need to adapt our solution accordingly. We are interested in guaranteeing optimal solutions after each graph change while retaining as much of the previous solution as possible. Even if the subgraph problem itself is polynomial-time computable, this multistage variant turns out to be NP-hard in most cases.