A $(1.5+ε)$-Approximation Algorithm for Weighted Connectivity Augmentation (2209.07860v1)

Abstract: Connectivity augmentation problems are among the most elementary questions in Network Design. Many of these problems admit natural $2$-approximation algorithms, often through various classic techniques, whereas it remains open whether approximation factors below $2$ can be achieved. One of the most basic examples thereof is the Weighted Connectivity Augmentation Problem (WCAP). In WCAP, one is given an undirected graph together with a set of additional weighted candidate edges, and the task is to find a cheapest set of candidate edges whose addition to the graph increases its edge-connectivity. We present a $(1.5+\varepsilon)$-approximation algorithm for WCAP, showing for the first time that factors below $2$ are achievable. On a high level, we design a well-chosen local search algorithm, inspired by recent advances for Weighted Tree Augmentation. To measure progress, we consider a directed weakening of WCAP and show that it has highly structured planar solutions. Interpreting a solution of the original problem as one of this directed weakening allows us to describe local exchange steps in a clean and algorithmically amenable way. Leveraging these insights, we show that we can efficiently search for good exchange steps within a component class for link sets that is closely related to bounded treewidth subgraphs of circle graphs. Moreover, we prove that an optimum solution can be decomposed into smaller components, at least one of which leads to a good local search step as long as we did not yet achieve the claimed approximation guarantee.