An eight-approximation algorithm for computing rooted three-vertex connected minimum steiner networks

For a given undirected (edge) weighted graph (G=(V,E)), a terminal set (S ⊂ V) and a root (r ∈ S), the rooted (k)-vertex connected minimum Steiner network (kVSMN_r) problem requires to construct a minimum-cost subgraph of (G) such that each terminal in S\{r} is (k)-vertex connected to (r). As an important problem in survivable network design, the (kVSMN _r) problem is known to be NP-hard even when (k=1). For (k=3) this paper presents a simple combinatorial eight-approximation algorithm, improving the known best ratio 14 of Nutov. Our algorithm constructs an approximate (3VSMN_r) through augmenting a two-vertex connected counterpart with additional edges of bounded cost to the optimal. We prove that the total cost of the added edges is at most six times of the optimal by showing that the edges in a (3VSMN_r) compose a subgraph containing our solution in such a way that each edge appears in the subgraph at most six times.