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.

翻译：给定定义在同一基础集上的两个拟阵$\mathcal{M}_1$和$\mathcal{M}_2$，拟阵交问题旨在寻找最大基数的公共独立集。在该问题的加权版本中，目标则是寻找最大权重的公共独立集。在不同计算模型下为该问题设计高效近似算法一直是研究热点。在众多模型中，无权版本与加权版本的最佳已知结果之间存在性能差距。本研究致力于弥合这一差距。我们的主要成果是一个归约方法，可将任意$α$近似的无权拟阵交算法转化为$α(1-\varepsilon)$近似的加权拟阵交算法，同时将算法运行时间增加$\log W$因子（其中$W$为权值跨度比）。该框架具有普适性，可推广至流式计算与单向通信复杂度等已深入研究的拟阵交计算场景。作为技术副产品，我们在这些模型中推导出了加权拟阵交问题的新结论。