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 unweighted 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 \(1/3 - \epsilon \), for any \(\epsilon > 0\). 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.

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