The Minimum Weighting Ratio Problem and Its Application in Chordal Graphs

TL;DR

This paper introduces a polynomial-time solution to the minimum weighting ratio problem for spanning trees and demonstrates its application in chordal graphs, advancing optimization techniques in graph theory.

Contribution

It presents a novel theorem that simplifies the minimum weighting ratio problem and applies it specifically to chordal graphs, providing new insights and methods.

Findings

The problem can be solved in polynomial time.

The approach is applicable to chordal graphs.

Theoretical foundation for ratio optimization in spanning trees.

Abstract

Constructing the maximum spanning tree $T$ of an edge-weighted connected graph $G$ is one of the important research topics in computer science and optimization, and the related research results have played an active role in practical applications. In this paper, we are concerned with the ratio of the weighted sum of a spanning tree $T$ of $G$ to the weighted sum of $G$, which we try to minimize. We propose an interesting theorem to simplify this problem and show that this optimal problem can be solved in polynomial time. Furthermore, we apply the optimal problem in chordal graphs.