A 4/3-approximation algorithm for finding a spanning tree to maximize its internal vertices

This paper focuses on finding a spanning tree of a graph to maximize the number of its internal vertices. We present an approximation algorithm for this problem which can achieve a performance ratio $\frac{4}{3}$ on undirected simple graphs. This improves upon the best known approximation algorithm with performance ratio $\frac{5}{3}$ before. Our algorithm benefits from a new observation for bounding the number of internal vertices of a spanning tree, which reveals that a spanning tree of an undirected simple graph has less internal vertices than the edges a maximum path-cycle cover of that graph has. We can also give an example to show that the performance ratio $\frac{4}{3}$ is actually tight for this algorithm. To decide how difficult it is for this problem to be approximated, we show that finding a spanning tree of an undirected simple graph to maximize its internal vertices is Max-SNP-Hard.Comment: 15 pages, 1 figur

Better approximation algorithms for maximum weight internal spanning trees in cubic graphs and claw-free graphs

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