A Lower Bound for the Rank of Matroid Intersection

TL;DR

This paper establishes a lower bound on the maximum size of common independent sets in matroid intersections, extending understanding beyond the well-studied case of two matroids to more complex intersections.

Contribution

It provides a lower bound estimate for the size of common independent sets in multiple matroid intersections and explores properties related to Edmonds' Min-max theorems.

Findings

Lower bound on the maximal cardinality of common independent sets

Properties of intersections of more than two matroids

Analogous results for Edmonds' Min-max theorems

Abstract

Matroid is a generalization of many fundamental objects in combinatorial mathematics , and matroid intersection problem is a classical subject in combinatorial optimization . However , only the intersection of two matroids are well understood . The solution of the intersection problem of more than three matroids is proved to be \textbf{NP-hard} . We will give a lower bound estimate on the maximal cardinality of the common independent sets in matroid intersections . We will also study some properties of the intersection of more than two matroids and deduce some analogous results for Edmonds' Min-max theorems for matroids intersection .