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 $$G =(V,E)$$ with non-negative costs $$c_{e}$$ for the edges $$e\in E$$ , a set of disjoint vertex sets $${\mathcal {V}}=\{V_{1},V_{2},\ldots ,V_{l}\}$$ and a parameter $$k\le l$$ . The goal is to find a minimum-cost edge set $$F\subseteq E$$ that connects at least k vertex sets in $${\mathcal {V}}$$ . Our main work is to construct an $$O(\sqrt{l})$$ -approximation algorithm for the k-GSF problem based on a greedy approach and an LP-rounding technique.