A 4 + ϵ approximation for k-connected subgraphs.

We obtain approximation ratio 4 + 2 ℓ ≈ 4 + 4 lg ⁡ 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 2 ℓ + 1 ≤ n. For large values of n this improves the ratio 6 of Cheriyan and Végh [4] when n ≥ k 3 (the case ℓ = 1). Our result implies an fpt-approximation ratio 4 + ϵ that matches (up to the "+ ϵ " 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 symmetric crossing supermodular biset function. [ABSTRACT FROM AUTHOR]