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We obtain approximation ratio $2(2+\frac{1}{\ell})$ for the (undirected)$k$-Connected Subgraph problem, where $\ell \approx \frac{1}{2} (\log_k n-1)$is the largest integer such that $2^{\ell-1} k^{2\ell+1} \leq n$. For largevalues of $n$ this improves the $6$-approximation of Cheriyan and V\'egh when$n =\Omega(k^3)$, which is the case $\ell=1$. For $k$ bounded by a constant weobtain ratio $4+\epsilon$. For large values of $n$ our ratio matches the bestknown ratio $4$ for the augmentation version of the problem, as well as thebest known ratios for $k=6,7$. Similar results are shown for the problem ofcovering an arbitrary crossing supermodular biset function.

A $(4+\epsilon)$-approximation for $k$-connected subgraphs