In this paper, we consider the tractability of the matroid intersection problem under the minimum rank oracle. In this model, we are given an oracle that takes as its input a set of elements, and returns as its output the minimum of the ranks of the given set in the two matroids. For the unweighted matroid intersection problem, we show how to construct a necessary part of the exchangeability graph, which enables us to emulate the standard augmenting path algorithm. Furthermore, we reformulate Edmonds' min-max theorem only using the minimum rank function, providing a new perspective on this result. For the weighted problem, the tractability is open in general. Nevertheless, we describe several special cases where tractability can be achieved, and we discuss potential approaches and the challenges encountered. In particular, we present a solution for the case where no circuit of one matroid is contained within a circuit of the other. Additionally, we propose a fixed-parameter tractable algorithm, parameterized by the maximum circuit size. We also show that a lexicographically maximal common independent set can be found by the same approach, which leads to at least $1/2$-approximation for finding a maximum-weight common independent set.

翻译：本文研究了在最小秩预言机模型下拟阵交问题的可解性。在此模型中，我们拥有一个预言机，其输入为元素集合，输出为该集合在两个拟阵中的秩的最小值。对于无权拟阵交问题，我们展示了如何构造可交换图的关键部分，从而能够模拟标准的增广路径算法。此外，我们仅使用最小秩函数重新表述了Edmonds的最小-最大定理，为该结果提供了新的视角。对于加权问题，其可解性在一般情况下尚未解决。尽管如此，我们描述了若干可实现可解性的特殊情形，并讨论了潜在方法及遇到的挑战。特别地，我们针对一个拟阵的任意回路均不包含于另一拟阵回路中的情形提出了解决方案。此外，我们提出了一种以最大回路规模为参数的固定参数可解算法。我们还证明了通过相同方法可以找到字典序最大的公共独立集，这为寻找最大权公共独立集提供了至少$1/2$的近似比。