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.

在本文中，我们介绍了k-广义Steiner森林（k -GSF）问题，它是k -Steiner森林问题和广义Steiner森林问题的自然概括。在这个问题中，我们得到了一个连通图\（G =（V，E）\），对于边\（e \ in E \）具有非负成本\（c_ {e} \），这是一个不相交的集合顶点集\（{\ mathcal {V}} = \ {V_ {1}，V_ {2}，\ ldots，V_ {l} \} \）和一个参数\（k \ le l \）。目标是找到一个最低成本的边缘集\（F \ subseteq E \），该边缘集至少连接\（{{mathcal {V}} \\}中的k个顶点集。我们的主要工作是基于贪婪方法和LP舍入技术构造一个针对k -GSF问题的\（O（\ sqrt {l}）\） -逼近算法。