An Improved Approximation Algorithm for the Minimum Cost Subset k-Connected Subgraph Problem

The minimum cost subset k-connected subgraph problem is a cornerstone problem in the area of network design with vertex connectivity requirements. In this problem, we are given a graph G=(V,E) with costs on edges and a set of terminals T⊆V. The goal is to find a minimum cost subgraph such that every pair of terminals are connected by k openly (vertex) disjoint paths. In this paper, we present an approximation algorithm for the subset k-connected subgraph problem which improves on the previous best approximation guarantee of O(k2logk) by Nutov (ACM Trans. Algorithms 9(1):1, 2012). Our approximation guarantee, α(|T|), depends upon the number of terminals: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha(|T|) = \begin{cases} O(k \log^2 k) & \mbox{if } 2k\le |T|< k^2\\ O(k \log k) & \mbox{if } |T|\ge k^2 \end{cases} $$\end{document} So, when the number of terminals is large enough, the approximation guarantee improves significantly. Moreover, we show that, given an approximation algorithm for |T|=k, we can obtain almost the same approximation guarantee for any instances with |T|>k. This suggests that the hardest instances of the problem are when |T|≈k.