An Approximation Algorithm for Maximum Internal Spanning Tree

TL;DR

This paper introduces a new approximation algorithm for the maximum internal spanning tree problem, improving the approximation ratio from 3/4 to 13/17 while maintaining polynomial time complexity.

Contribution

It presents a refined approximation algorithm that explores deeper problem structures, achieving a better ratio for MIST without increasing computational complexity.

Findings

Achieves a 13/17 approximation ratio for MIST

Maintains polynomial time complexity

Identifies deeper structural properties of the problem

Abstract

Given a graph G, the {\em maximum internal spanning tree problem} (MIST for short) asks for computing a spanning tree T of G such that the number of internal vertices in T is maximized. MIST has possible applications in the design of cost-efficient communication networks and water supply networks and hence has been extensively studied in the literature. MIST is NP-hard and hence a number of polynomial-time approximation algorithms have been designed for MIST in the literature. The previously best polynomial-time approximation algorithm for MIST achieves a ratio of 3/4. In this paper, we first design a simpler algorithm that achieves the same ratio and the same time complexity as the previous best. We then refine the algorithm into a new approximation algorithm that achieves a better ratio (namely, 13/17) with the same time complexity. Our new algorithm explores much deeper structure of the problem than the previous best. The discovered structure may be used to design even better approximation or parameterized algorithms for the problem in the future.