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The minimum-cost subset $k$-connected subgraph problem is a cornerstoneproblem in the area of network design with vertex connectivity requirements. Inthis problem, we are given a graph $G=(V,E)$ with costs on edges and a set ofterminals $T$. The goal is to find a minimum cost subgraph such that every pairof terminals are connected by $k$ openly (vertex) disjoint paths. In thispaper, we present an approximation algorithm for the subset $k$-connectedsubgraph problem which improves on the previous best approximation guarantee of$O(k^2\log{k})$ by Nutov (FOCS 2009). Our approximation guarantee,$\alpha(|T|)$, depends upon the number of terminals: [\alpha(|T|) \ \ =\ \O(|T|^2) & if |T| < 2k O(k \log^2 k) & if 2k\le |T| < k^2 O(k \log k) & if |T|\ge k^2]  So, when the number of terminals is {\em large enough}, the approximationguarantee improves significantly. Moreover, we show that, given anapproximation algorithm for $|T|=k$, we can obtain almost the sameapproximation guarantee for any instances with $|T|> k$. This suggests that thehardest instances of the problem are when $|T|\approx k$.

An improved approximation algorithm for the minimum-cost subset k-connected subgraph problem