Consider a graph with nonnegative node weight. A vertex subset is called a CDS (connected dominating set) if every other node has at least one neighbor in the subset and the subset induces a connected subgraph. Furthermore, if every other node has at least $m$ neighbors in the subset, then the node subset is called a $(1,m)$CDS. The minimum-weight $(1,m)$CDS problem aims at finding a $(1,m)$CDS with minimum total node weight. In this paper, we present a new polynomial-time approximation algorithm for this problem with approximation ratio $2H(\delta_{\max}+m-1)$, where $\delta_{\max}$ is the maximum degree of the given graph and $H(\cdot)$ is the Harmonic function, i.e., $H(k)=\sum_{i=1}^k \frac{1}{i}$.

翻译：考虑一个具有非负节点权重的图。如果一个节点子集使得所有其他节点至少有一个邻居在该子集中，并且该子集诱导出一个连通子图，则称该子集为CDS（连通支配集）。此外，如果每个其他节点在该子集中至少有$m$个邻居，则称该节点子集为$(1,m)$CDS。最小权重$(1,m)$CDS问题旨在寻找总节点权重最小的$(1,m)$CDS。本文针对该问题提出了一种新的多项式时间近似算法，其近似比为$2H(\delta_{\max}+m-1)$，其中$\delta_{\max}$为给定图的最大度，$H(\cdot)$为调和函数，即$H(k)=\sum_{i=1}^k \frac{1}{i}$。