Approximation algorithms for the maximum weight internal spanning tree problem

Given a vertex-weighted connected graph $G = (V, E)$, the maximum weight internal spanning tree (MwIST for short) problem asks for a spanning tree $T$ of $G$ such that the total weight of the internal vertices in $T$ is maximized. The un-weighted variant, denoted as MIST, is NP-hard and APX-hard, and the currently best approximation algorithm has a proven performance ratio $13/17$. The currently best approximation algorithm for MwIST only has a performance ratio $1/3 - ε$, for any $ε> 0$. In this paper, we present a simple algorithm based on a novel relationship between MwIST and the maximum weight matching, and show that it achieves a better approximation ratio of $1/2$. When restricted to claw-free graphs, a special case been previously studied, we design a $7/12$-approximation algorithm.