Given two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ over the same $n$-element ground set, the matroid intersection problem is to find a largest common independent set, whose size we denote by $r$. We present a simple and generic auction algorithm that reduces $(1-\varepsilon)$-approximate matroid intersection to roughly $1/\varepsilon^2$ rounds of the easier problem of finding a maximum-weight basis of a single matroid. Plugging in known primitives for this subproblem, we obtain both simpler and improved algorithms in two models of computation, including: * The first near-linear time/independence-query $(1-\varepsilon)$-approximation algorithm for matroid intersection. Our randomized algorithm uses $\tilde{O}(n/\varepsilon + r/\varepsilon^5)$ independence queries, improving upon the previous $\tilde{O}(n/\varepsilon + r\sqrt{r}/{\varepsilon^3})$ bound of Quanrud (2024). * The first sublinear exact parallel algorithms for weighted matroid intersection, using $O(n^{2/3})$ rounds of rank queries or $O(n^{5/6})$ rounds of independence queries. For the unweighted case, our results improve upon the previous $O(n^{3/4})$-round rank-query and $O(n^{7/8})$-round independence-query algorithms of Blikstad (2022).

翻译：给定定义在相同 $n$ 元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一个最大的公共独立集，其大小记为 $r$。本文提出一种简洁且通用的拍卖算法，将 $(1-\varepsilon)$-近似拟阵交问题约化为约 $1/\varepsilon^2$ 轮更简单的单拟阵最大权基问题。通过引入该子问题的已知原语，我们在两种计算模型中获得了更简洁且改进的算法，包括：* 首个拟阵交问题的近线性时间/独立性查询 $(1-\varepsilon)$-近似算法。我们的随机算法使用 $\tilde{O}(n/\varepsilon + r/\varepsilon^5)$ 次独立性查询，改进了 Quanrud (2024) 先前提出的 $\tilde{O}(n/\varepsilon + r\sqrt{r}/{\varepsilon^3})$ 界。* 首个加权拟阵交问题的亚线性精确并行算法，使用 $O(n^{2/3})$ 轮秩查询或 $O(n^{5/6})$ 轮独立性查询。对于无权情形，我们的结果改进了 Blikstad (2022) 先前提出的 $O(n^{3/4})$ 轮秩查询与 $O(n^{7/8})$ 轮独立性查询算法。