Efficiently Coloring the Intersection of a General Matroid and Partition Matroids

TL;DR

This paper introduces a polynomial-time algorithm for coloring the intersection of a general matroid with multiple partition matroids, achieving an approximation ratio close to optimal, which was previously unknown.

Contribution

It provides the first polynomial-time O(1)-approximation algorithm for matroid intersection coloring involving a general matroid, expanding the scope of efficient coloring algorithms.

Findings

First polynomial-time O(1)-approximation for general matroid intersection coloring.

Algorithm extends to standard combinatorial matroids with similar approximation guarantees.

Achieves near-optimal coloring bounds in polynomial time.

Abstract

This paper shows a polynomial-time algorithm, that given a general matroid $M_1 = (X, \mathcal{I}_1)$ and $k-1$ partition matroids $ M_2, \ldots, M_k$, produces a coloring of the intersection $M = \cap_{i=1}^k M_i$ using at most $1+\sum_{i=1}^k \left(\chi(M_i) -1\right)$ colors. This is the first polynomial-time $O(1)$-approximation algorithm for matroid intersection coloring where one of the matroids may be a general matroid. Leveraging the fact that all of the standard combinatorial matroids reduce to partition matroids at a loss of a factor of two in the chromatic number, this algorithm also yields a polynomial-time $O(1)$-approximation algorithm for matroid intersection coloring in the case where each of the matroids $ M_2, \ldots, M_k$ are one of the standard combinatorial types.