A General Approach to Approximate Multistage Subgraph Problems

Abstract

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/ √ 2χ)-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 χ, 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.

Approximating Multistage Matching Problems

“…Furthermore, it turns out that the techniques of Algorithms 1 and 2 for MIM can be used as a building block to approximate a broad set of related problems, so-called Multistage Subgraph Problems [12]. We are not aware of any way to circumvent this problem.…”