Approximation algorithms for the maximum weight internal spanning tree problem

TL;DR

This paper introduces a new approximation algorithm for the maximum weight internal spanning tree problem, improving the performance ratio from 1/3 to 1/2, and offers a specialized solution for claw-free graphs.

Contribution

It presents a simple, effective approximation algorithm for MwIST with a 1/2 ratio, based on a novel link to maximum weight matching, and extends to claw-free graphs.

Findings

Achieves a 1/2 approximation ratio for MwIST

Provides a 7/12 approximation for claw-free graphs

Improves upon the previous best approximation ratios

Abstract

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 $1/3 - \epsilon$, 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.