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

TL;DR

This paper introduces a new approximation algorithm with a 4/3 ratio for maximizing internal vertices in a spanning tree, improving previous ratios and proving the problem's computational hardness.

Contribution

It presents a novel 4/3-approximation algorithm for the problem, improving the previous 5/3 ratio, and establishes the problem's Max-SNP-Hardness.

Findings

The algorithm achieves a 4/3 performance ratio.

The problem is Max-SNP-Hard.

The algorithm's ratio is proven tight.

Abstract

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.