A 4+ε approximation for k-connected subgraphs

We obtain approximation ratio 4 + 2/l approximate to 4 + 4lg k/lg n-lg k for the (undirected) k-CONNECTED SUBGRAPH problem, where l = [lg n-lg k+1/2 lg k+1] is the largest integer such that 2(l-1)k(2l+1)<= n. For large values of n this improves the ratio 6 of Cheriyan and Vegh [4] when n >= k(3) (the case l = 1). Our result implies an fpt-approximation ratio 4 + epsilon that matches (up to the "+epsilon" term) the best known ratio 4 for k = 6, 7 for both the general and the easier augmentation versions of the problem. Similar results are shown for the problem of covering an arbitrary crossing supermodular biset function.