Matroid intersection is one of the most powerful frameworks of matroid theory that generalizes various problems in combinatorial optimization. Edmonds' fundamental theorem provides a min-max characterization for the unweighted setting, while Frank's weight-splitting theorem provides one for the weighted case. Several efficient algorithms were developed for these problems, all relying on the usage of one of the conventional oracles for both matroids. In the present paper, we consider the tractability of the matroid intersection problem under restricted oracles. In particular, we focus on the rank sum, common independence, and maximum rank oracles. We give a strongly polynomial-time algorithm for weighted matroid intersection under the rank sum oracle. In the common independence oracle model, we prove that the unweighted matroid intersection problem is tractable when one of the matroids is a partition matroid, and that even the weighted case is solvable when one of the matroids is an elementary split matroid. Finally, we show that the common independence and maximum rank oracles together are strong enough to realize the steps of our algorithm under the rank sum oracle.

翻译：拟阵交是拟阵理论中最强大的框架之一，它概括了组合优化中的各种问题。Edmonds的基本定理为无权重情形提供了极小-极大刻画，而Frank的权重分割定理则为加权情形提供了相应刻画。针对这些问题，已有多种高效算法被提出，这些算法均依赖于对两个拟阵使用传统预言机之一。本文研究了受限预言机下拟阵交问题的可解性，特别关注秩和预言机、公共独立预言机和最大秩预言机。我们给出了秩和预言机下加权拟阵交的强多项式时间算法。在公共独立预言机模型下，我们证明当其中一个拟阵为分割拟阵时，无权重拟阵交问题是可解的；而当其中一个拟阵为初等分裂拟阵时，即使加权情形也是可解的。最后，我们证明公共独立预言机和最大秩预言机共同足以实现我们在秩和预言机下的算法步骤。