Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

Abstract

Given a k-vertex-connected graph G and a set S of extra edges (links), the goal of the kvertex-connectivity augmentation problem is to find a set S ⊆ S of minimum size such that adding S to G makes it (k + 1)-vertex-connected. Unlike the edge-connectivity augmentation problem, research for the vertex-connectivity version has been sparse.In this work we present the first polynomial time approximation algorithm that improves the known ratio of 2 for 2-vertex-connectivity augmentation, for the case in which G is a cycle. This is the first step for attacking the more general problem of augmenting a 2-connected graph.Our algorithm is based on local search and attains an approximation ratio of 1.8704. To derive it, we prove novel results on the structure of minimal solutions. * Full version of the extended abstract accepted at WAOA 2021. In that extended abstract an approximation ratio of 1.8703 was claimed, but it has been corrected to 1.8704 in this version. We apologize for the inconvenience.