An Efficient Reduction of a Gammoid to a Partition Matroid

TL;DR

This paper presents a polynomial-time algorithm to reduce a k-colorable gammoid to a (2k-2)-colorable partition matroid, providing a tight bound and improving algorithms for related coloring problems.

Contribution

It introduces a tight polynomial-time reduction from gammoids to partition matroids, enabling better approximation algorithms for matroid intersection problems.

Findings

Reduction is tight, cannot be improved below (2k-2)

Enables polynomial-time algorithms with improved approximation ratios

Applicable to coloring and list coloring problems in matroid intersections

Abstract

Our main contribution is a polynomial-time algorithm to reduce a $k$-colorable gammoid to a $(2k-2)$-colorable partition matroid. It is known that there are gammoids that can not be reduced to any $(2k-3)$-colorable partition matroid, so this result is tight. We then discuss how such a reduction can be used to obtain polynomial-time algorithms with better approximation ratios for various natural problems related to coloring and list coloring the intersection of matroids.