Given a connected vertex-weighted graph $G$, the maximum weight internal spanning tree (MaxwIST) problem asks for a spanning tree of $G$ that maximizes the total weight of internal nodes. This problem is NP-hard and APX-hard, with the currently best known approximation factor $1/2$ (Chen et al., Algorithmica 2019). For the case of claw-free graphs, Chen et al. present an involved approximation algorithm with approximation factor $7/12$. They asked whether it is possible to improve these ratios, in particular for claw-free graphs and cubic graphs. We improve the approximation factors for the MaxwIST problem in cubic graphs and claw-free graphs. For cubic graphs we present an algorithm that computes a spanning tree whose total weight of internal vertices is at least $\frac{3}{4}-\frac{3}{n}$ times the total weight of all vertices, where $n$ is the number of vertices of $G$. This ratio is almost tight for large values of $n$. For claw-free graphs of degree at least three, we present an algorithm that computes a spanning tree whose total internal weight is at least $\frac{3}{5}-\frac{1}{n}$ times the total vertex weight. The degree constraint is necessary as this ratio may not be achievable if we allow vertices of degree less than three. With the above ratios, we immediately obtain better approximation algorithms with factors $\frac{3}{4}-\epsilon$ and $\frac{3}{5}-\epsilon$ for the MaxwIST problem in cubic graphs and claw-free graphs of degree at least three, for any $\epsilon>0$. In addition to improving the approximation factors, the new algorithms are relatively short compared to that of Chen et al.. The new algorithms are fairly simple, and employ a variant of the depth-first search algorithm that selects a relatively-large-weight vertex in every branching step. Moreover, the new algorithms take linear time while previous algorithms for similar problem instances are super-linear.

中文翻译：

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三次图和无爪图中最大权重内部生成树的更好近似算法

给定一个连接的顶点加权图 $G$，最大权重内部生成树 (MaxwIST) 问题要求生成 $G$ 的生成树，以最大化内部节点的总权重。这个问题是 NP-hard 和 APX-hard，目前最著名的近似因子是 $1/2$（Chen 等人，Algorithmica 2019）。对于无爪图的情况，Chen 等人。提出一个近似因子为 $7/12$ 的近似算法。他们询问是否有可能提高这些比率，特别是对于无爪图和三次图。我们改进了三次图和无爪图中 MaxwIST 问题的近似因子。对于三次图，我们提出了一种计算生成树的算法，其内部顶点的总权重至少为 $\frac{3}{4}-\frac{3}{n}$ 乘以所有顶点的总权重，其中 $n$ 是 $G$ 的顶点数。对于 $n$ 的大值，这个比率几乎很紧。对于度数至少为 3 的无爪图，我们提出了一种计算生成树的算法，该生成树的总内部权重至少为 $\frac{3}{5}-\frac{1}{n}$ 乘以总顶点重量。度数约束是必要的，因为如果我们允许度数小于 3 的顶点，则此比率可能无法实现。有了上述比率，我们立即获得了更好的近似算法，其中因子 $\frac{3}{4}-\epsilon$ 和 $\frac{3}{5}-\epsilon$ 用于三次图中的 MaxwIST 问题和爪-对于任何 $\epsilon>0$，度数至少为 3 的自由图。除了改进近似因子外，新算法与 Chen 等人的算法相比相对较短。 新算法相当简单，并采用深度优先搜索算法的变体，在每个分支步骤中选择一个相对较大的权重顶点。此外，新算法需要线性时间，而以前针对类似问题实例的算法是超线性的。