An Improved Approximation Algorithm for Minimum-Cost Subset k-Connectivity

Abstract

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. 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(k ²logk) by Nutov (FOCS 2009). Our approximation guarantee, α(|T |), depends upon the number of terminals:

$$ \alpha(|T|) \ \ =\ \ \begin{cases} O(|T|^2) &\mbox{if } |T| < 2k\\ O(k \log^2 k) & \mbox{if } 2k\le |T| < k^2\\ O(k \log k) & \mbox{if } |T| \ge k^2 \end{cases} $$

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.