Vertex covering with capacitated trees

Abstract

The covering of a graph with (possibly disjoint) connected subgraphs is a fundamental problem in graph theory. In this paper, we study a version to cover a graph's vertices by connected subgraphs subject to lower and upper weight bounds, and propose a column generation approach to dynamically generate feasible and promising subgraphs. Our focus is on the solution of the pricing problem which turns out to be a variant of the NP‐hard Maximum Weight Connected Subgraph Problem. We compare different formulations to handle connectivity, and find that a single‐commodity flow formulation performs best. This is notable since the respective literature seems to have widely dismissed this formulation. We improve it to a new coarse‐to‐fine flow formulation that is theoretically and computationally superior, especially for large instances with many vertices of degree 2 like highway networks, where it provides a speed‐up factor of 5 over the non‐flow‐based formulations. We also propose a preprocessing method that exploits a median property of weight‐constrained subgraphs, a primal heuristic, and a local search heuristic. In an extensive computational study we evaluate the presented connectivity formulations on different classes of instances, and demonstrate the effectiveness of the proposed enhancements. Their speed‐ups essentially multiply to an overall factor of well over 10. Overall, our approach allows the reliable solution of instances with several hundreds of vertices in a few minutes. These findings are further corroborated in a comparison to existing districting models on a set of test instances from the literature.