Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

TL;DR

This paper introduces a polynomial-time approximation algorithm with a ratio of 1.8704 for augmenting a cycle to increase its vertex connectivity from 2 to 3, advancing the understanding of vertex-connectivity augmentation.

Contribution

It presents the first improved approximation algorithm for vertex-connectivity augmentation on cycles, using local search and novel structural results.

Findings

Achieved an approximation ratio of 1.8704 for the problem.

First polynomial-time algorithm with ratio better than 2 for this problem.

Provides new insights into the structure of minimal solutions.

Abstract

Given a $k$-vertex-connected graph $G$ and a set $S$ of extra edges (links), the goal of the $k$-vertex-connectivity augmentation problem is to find a set $S' \subseteq S$ 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.