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 [SICOMP 1986] 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, where $k' = \min\{k, p\}$. This query complexity is $\tilde{O}(n^{7/3})$ when $k \leq n$, and this improves upon the one of 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 [2019] and Chakrabarty-Lee-Sidford-Singla-Wong [FOCS 2019] and a careful analysis of the independence oracle query complexity. 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 $V$（含 $n$ 个元素）上的 $k$ 个拟阵 $\mathcal{M}_1 = (V, \mathcal{I}_1), \dots , \mathcal{M}_k = (V, \mathcal{I}_k)$，需寻找最大可能基数的可划分集 $S \subseteq V$，记该基数为 $p$。其中，集合 $S \subseteq V$ 被称为可划分集，若存在 $S$ 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 [SICOMP 1986] 提出了一种使用 $O(n p^{3/2} + k n)$ 次独立性子例查询的拟阵划分算法，此为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使用 $\tilde{O}(k'^{1/3} n p + k n)$ 次独立性子例查询的拟阵划分算法，其中 $k' = \min\{k, p\}$。当 $k \leq n$ 时，此查询复杂度为 $\tilde{O}(n^{7/3})$，优于先前Cunningham算法。为此，我们提出了一种新方法——\emph{边回收增广}，该方法通过以下新思路实现：高效利用Nguyen [2019] 与 Chakrabarty-Lee-Sidford-Singla-Wong [FOCS 2019] 的二分搜索技术，以及细致分析独立性子例查询复杂度。由于参数 $k$ 的存在，我们的分析与拟阵交算法显著不同。我们还提出了一种使用 $\tilde{O}((n + k) \sqrt{p})$ 次秩子例查询的拟阵划分算法。