Efficiently Coloring the Intersection of a General Matroid and Partition Matroids

2025-08-26

AI Summary

This paper studies edge coloring on the intersection of a general matroid and multiple partition matroids—a problem proven to be NP-hard. We propose the first polynomial-time $O(1)$-approximation algorithm for the setting where one matroid is general and the others are partition matroids. Our method embeds the general matroid into a partition matroid structure via combinatorial reduction, then designs an efficient algorithm combining matroid basis exchange with greedy coloring, ensuring feasibility under the independence constraints of the matroid intersection. The algorithm uses at most $1 + sum_i (chi(M_i) - 1)$ colors—significantly improving upon prior algorithms restricted to intersections of solely partition matroids. This is the first constant-factor approximation algorithm for matroid intersection coloring that accommodates a general matroid, thereby extending both the theoretical scope and practical applicability of matroid coloring.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $1+sum_{i=1}^k left(χ(M_i) -1 ight)$ 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.

Problem

• Develops polynomial-time algorithm for matroid intersection coloring
• Approximates coloring using O(1) colors for general matroids
• Extends to standard combinatorial matroids with constant approximation

Innovation

• Polynomial-time algorithm for matroid intersection coloring
• Approximates coloring with O(1) approximation factor
• Handles general matroid plus partition matroids combination