Min-Cost 2-Connected Subgraphs With k Terminals

TL;DR

This paper presents an O(log n log k) approximation algorithm for the k-2VC problem, which seeks the minimum-cost 2-vertex-connected subgraph containing at least k vertices, extending previous work on related connectivity problems.

Contribution

It introduces the first approximation algorithm for the general k-2VC problem, improving the understanding of cost-efficient 2-vertex-connected subgraphs.

Findings

Provides an O(log n log k) approximation algorithm.

Extends the approximation techniques from k-2EC to k-2VC.

Addresses a previously open problem in graph connectivity optimization.

Abstract

In the k-2VC problem, we are given an undirected graph G with edge costs and an integer k; the goal is to find a minimum-cost 2-vertex-connected subgraph of G containing at least k vertices. A slightly more general version is obtained if the input also specifies a subset S \subseteq V of terminals and the goal is to find a subgraph containing at least k terminals. Closely related to the k-2VC problem, and in fact a special case of it, is the k-2EC problem, in which the goal is to find a minimum-cost 2-edge-connected subgraph containing k vertices. The k-2EC problem was introduced by Lau et al., who also gave a poly-logarithmic approximation for it. No previous approximation algorithm was known for the more general k-2VC problem. We describe an O(\log n \log k) approximation for the k-2VC problem.