A General Approach to Approximate Multistage Subgraph Problems

TL;DR

This paper introduces a framework for approximating multistage subgraph problems, providing guarantees based on graph similarity measures and extending solutions to multiple stages for various classical graph problems.

Contribution

It presents a novel approximation algorithm for 2-stage multistage subgraph problems and extends the approach to multiple stages, applicable to many classical graph problems.

Findings

Provides a $(1/\sqrt{2 ext{chi}})$-approximation for 2-stage MSPs with intersection similarity.

Extends approximation methods to unrestricted stages using solutions from constant-stage algorithms.

Applies the framework to problems like Perfect Matching, Shortest Path, and Max Cut on various graph classes.

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 $(1/\sqrt{2\chi})$-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.