Approximation Algorithms for Matroid-Intersection Coloring with Applications to Rota's Basis Conjecture

TL;DR

This paper introduces polynomial-time algorithms for matroid intersection coloring, providing constructive approximations that depend only on the number of matroids, and offers an FPRAS for large chromatic numbers, advancing towards Rota's Basis Conjecture.

Contribution

It presents the first polynomial-time algorithms with approximation ratios depending only on the number of matroids, and introduces a novel matroidal structure called flexible decomposition.

Findings

Achieves a 2-approximation for two matroids, matching the existential bound.

Provides a $(k^2 - k) ext{-approximation}$ for $k$ matroids, the first $O(1)$-approximation for constant $k$.

Develops an FPRAS for coloring the intersection of two matroids with large $ ext{chromatic number}$.

Abstract

We study algorithmic matroid intersection coloring. Given $k$ matroids on a common ground set $U$ of $n$ elements, the goal is to partition $U$ into the fewest number of color classes, where each color class is independent in all matroids. It is known that $2\chi_{\max}$ colors suffice to color the intersection of two matroids, $(2k-1)\chi_{\max}$ colors suffice for general $k$, where $\chi_{\max}$ is the maximum chromatic number of the individual matroids. However, these results are non-constructive, leveraging techniques such as topological Hall's theorem and Sperner's Lemma. We provide the first polynomial-time algorithms to color two or more general matroids where the approximation ratio depends only on $k$ and, in particular, is independent of $n$. For two matroids, we constructively match the $2\chi_{\max}$ existential bound, yielding a 2-approximation for the Matroid Intersection Coloring problem. For $k$ matroids we achieve a $(k^2-k)\chi_{\max}$ coloring, which is the first $O(1)$-approximation for constant $k$. Our approach introduces a novel matroidal structure we call a \emph{flexible decomposition}. We use this to formally reduce general matroid intersection coloring to graph coloring while avoiding the limitations of partition reduction techniques, and without relying on non-constructive topological machinery. Furthermore, we give a \emph{fully polynomial randomized approximation scheme} (FPRAS) for coloring the intersection of two matroids when $\chi_{\max}$ is large. This yields the first polynomial-time constructive algorithm for an asymptotic variant of Rota's Basis Conjecture. This constructivizes Montgomery and Sauermann's recent asymptotic breakthrough and generalizes it to arbitrary matroids.