A Generalized-Polymatroid Approach to Disjoint Common Independent Sets in Two Matroids

TL;DR

This paper introduces a generalized-polymatroid approach to solve partitioning problems in two matroids, providing new proofs and extending results to broader classes of matroids, with some limitations.

Contribution

It develops a unified framework using generalized-polymatroids for partitioning in two matroids and extends known results to new matroid classes.

Findings

Solution exists for laminar matroids intersection

Solution exists for matroids without $(k+1)$-spanned elements

Counterexample with transversal matroid

Abstract

In this paper, we investigate the classes of matroid intersection admitting a solution for the problem of partitioning the ground set $E$ into $k$ common independent sets, where $E$ can be partitioned into $k$ independent sets in each of the two matroids. For this problem, we present a new approach building upon the generalized-polymatroid intersection theorem. We exhibit that this approach offers alternative proofs and unified understandings of previous results showing that the problem has a solution for the intersection of two laminar matroids and that of two matroids without $(k+1)$-spanned elements. Moreover, we newly show that the intersection of a laminar matroid and a matroid without $(k+1)$-spanned elements admits a solution. We also construct an example of a transversal matroid which is incompatible with the generalized-polymatroid approach.