A Weighted-to-Unweighted Reduction for Matroid Intersection

TL;DR

This paper introduces a reduction technique that transforms unweighted matroid intersection algorithms into weighted ones with minimal loss in approximation quality, improving efficiency across various computational models.

Contribution

The authors present a novel reduction method that converts any $ ext{α}$-approximate unweighted matroid intersection algorithm into an $( ext{α}(1- ext{ε}))$-approximate weighted version, with only a logarithmic increase in runtime.

Findings

Achieves near-optimal approximation for weighted matroid intersection

Applies to streaming and communication complexity models

Derives new results for weighted matroid intersection in these models

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 $\alpha$-approximate unweighted matroid intersection algorithm into an $\alpha(1-\varepsilon)$-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.