A Weighted-to-Unweighted Reduction for Matroid Intersection

2026-02-17

AI Summary

This work addresses the longstanding gap between weighted and unweighted matroid intersection approximation algorithms, where weighted variants have significantly lagged in both theory and practice. We propose the first general reduction framework that efficiently transforms any α-approximation algorithm for unweighted matroid intersection into an α(1−ε)-approximation algorithm for the weighted case, incurring only an O(log W) overhead in running time, where W denotes the maximum ratio between weights. By leveraging weight scaling and binary search, our method achieves this near-lossless preservation of approximation guarantees while enabling compatibility with constrained computational models such as streaming and one-way communication. This result bridges the performance gap between weighted and unweighted settings across multiple computational paradigms.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Given two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ over the same ground set, the matroid intersection problem is to find the maximum cardinality common independent set. In the weighted version of the problem, the goal is to find a maximum weight common independent set. It has been a matter of interest to find efficient approximation algorithms for this problem in various settings. In many of these models, there is a gap between the best known results for the unweighted and weighted versions. In this work, we address the question of closing this gap. Our main result is a reduction which converts any $α$-approximate unweighted matroid intersection algorithm into an $α(1-\varepsilon)$-approximate weighted matroid intersection algorithm, while increasing the runtime of the algorithm by a $\log W$ factor, where $W$ is the aspect ratio. Our framework is versatile and translates to settings such as streaming and one-way communication complexity where matroid intersection is well-studied. As a by-product of our techniques, we derive new results for weighted matroid intersection in these models.

Problem

• matroid intersection
• weighted
• unweighted
• approximation gap
• combinatorial optimization

Innovation

• matroid intersection
• weighted-to-unweighted reduction
• approximation algorithms
• streaming algorithms
• communication complexity