In the matroid partitioning problem, we are given $k$ matroids $\mathcal{M}_1 = (V, \mathcal{I}_1), \dots , \mathcal{M}_k = (V, \mathcal{I}_k)$ defined over a common ground set $V$ of $n$ elements, and we need to find a partitionable set $S \subseteq V$ of largest possible cardinality, denoted by $p$. Here, a set $S \subseteq V$ is called partitionable if there exists a partition $(S_1, \dots , S_k)$ of $S$ with $S_i \in \mathcal{I}_i$ for $i = 1, \ldots, k$. In 1986, Cunningham presented a matroid partition algorithm that uses $O(n p^{3/2} + k n)$ independence oracle queries, which was the previously known best algorithm. This query complexity is $O(n^{5/2})$ when $k \leq n$. Our main result is to present a matroid partition algorithm that uses $\tilde{O}(k^{1/3} n p + k n)$ independence oracle queries, which is $\tilde{O}(n^{7/3})$ when $k \leq n$. This improves upon previous Cunningham's algorithm. To obtain this, we present a new approach \emph{edge recycling augmentation}, which can be attained through new ideas: an efficient utilization of the binary search technique by Nguyen and Chakrabarty-Lee-Sidford-Singla-Wong and a careful analysis of the number of independence oracle queries. Our analysis differs significantly from the one for matroid intersection algorithms, because of the parameter $k$. We also present a matroid partition algorithm that uses $\tilde{O}((n + k) \sqrt{p})$ rank oracle queries.

翻译：在拟阵划分问题中，给定定义在公共基集 $V$（包含 $n$ 个元素）上的 $k$ 个拟阵 $\mathcal{M}_1 = (V, \mathcal{I}_1), \dots , \mathcal{M}_k = (V, \mathcal{I}_k)$，需要找到最大可能基数（记为 $p$）的可划分集 $S \subseteq V$。这里，集合 $S \subseteq V$ 称为可划分的，如果存在 $S$ 的一个划分 $(S_1, \dots , S_k)$，使得对 $i = 1, \ldots, k$ 有 $S_i \in \mathcal{I}_i$。1986年，Cunningham 提出了一种拟阵划分算法，该算法使用了 $O(n p^{3/2} + k n)$ 次独立性谕示查询，这是此前已知的最佳算法。当 $k \leq n$ 时，该查询复杂度为 $O(n^{5/2})$。我们的主要结果是提出一种拟阵划分算法，该算法使用了 $\tilde{O}(k^{1/3} n p + k n)$ 次独立性谕示查询，当 $k \leq n$ 时复杂度为 $\tilde{O}(n^{7/3})$。这改进了此前 Cunningham 的算法。为实现这一目标，我们提出了一种新方法——\emph{边回收增广}，该方法可通过以下新思路实现：有效利用 Nguyen 和 Chakrabarty-Lee-Sidford-Singla-Wong 的二分搜索技术，以及对独立性谕示查询次数的细致分析。由于参数 $k$ 的存在，我们的分析与拟阵交算法中的分析有显著不同。我们还提出了一种使用 $\tilde{O}((n + k) \sqrt{p})$ 次秩谕示查询的拟阵划分算法。