An approximation algorithm for the k-generalized Steiner forest problem

In this paper, we introduce the k -generalized Steiner forest ( k -GSF) problem, which is a natural generalization of the k -Steiner forest problem and the generalized Steiner forest problem. In this problem, we are given a connected graph $$G =(V,E)$$ G = ( V , E ) with non-negative costs $$c_{e}$$ c e for the edges $$e\in E$$ e ∈ E , a set of disjoint vertex sets $${\mathcal {V}}=\{V_{1},V_{2},\ldots ,V_{l}\}$$ V = { V 1 , V 2 , … , V l } and a parameter $$k\le l$$ k ≤ l . The goal is to find a minimum-cost edge set $$F\subseteq E$$ F ⊆ E that connects at least k vertex sets in $${\mathcal {V}}$$ V . Our main work is to construct an $$O(\sqrt{l})$$ O ( l ) -approximation algorithm for the k -GSF problem based on a greedy approach and an LP-rounding technique.