Matroid reinforcement and sparsification

TL;DR

This paper explores methods to transform matroids into homogeneous ones through weight adjustments, providing algorithms for optimal reinforcement and sparsification, with applications to graph spanning trees.

Contribution

It introduces new algorithms for matroid transformation via weight modifications, extending previous work on graph reinforcement to a broader matroid setting.

Findings

Algorithms for matroid reinforcement and sparsification

Efficient solutions for transforming matroids into homogeneous forms

Applications to optimizing spanning trees in graphs

Abstract

Homogeneous matroids are characterized by the property that strength equals fractional arboricity, and arise in the study of base modulus [22]. For graphic matroids, Cunningham [9] provided efficient algorithms for calculating graph strength, and also for determining minimum cost reinforcement to achieve a desired strength. This paper extends this latter problem by focusing on two optimal strategies for transforming a matroid into a homogeneous one, by either increasing or decreasing element weights. As an application to graphs, we give algorithms to solve this problem in the context of spanning trees.