Given two matroids $\mathcal{M}_1 = (V, \mathcal{I}_1)$ and $\mathcal{M}_2 = (V, \mathcal{I}_2)$ over an $n$-element integer-weighted ground set $V$, the weighted matroid intersection problem aims to find a common independent set $S^{*} \in \mathcal{I}_1 \cap \mathcal{I}_2$ maximizing the weight of $S^{*}$. In this paper, we present a simple deterministic algorithm for weighted matroid intersection using $\tilde{O}(nr^{3/4}\log{W})$ rank queries, where $r$ is the size of the largest intersection of $\mathcal{M}_1$ and $\mathcal{M}_2$ and $W$ is the maximum weight. This improves upon the best previously known $\tilde{O}(nr\log{W})$ algorithm given by Lee, Sidford, and Wong [FOCS'15], and is the first subquadratic algorithm for polynomially-bounded weights under the standard independence or rank oracle models. The main contribution of this paper is an efficient algorithm that computes shortest-path trees in weighted exchange graphs.

翻译：给定两个定义在$n$元整数权基集$V$上的拟阵$\mathcal{M}_1 = (V, \mathcal{I}_1)$和$\mathcal{M}_2 = (V, \mathcal{I}_2)$，加权拟阵交问题旨在寻找一个共同独立集$S^{*} \in \mathcal{I}_1 \cap \mathcal{I}_2$，使得$S^{*}$的权重最大化。本文提出了一种简单的确定性算法，通过$\tilde{O}(nr^{3/4}\log{W})$次秩查询解决加权拟阵交问题，其中$r$是$\mathcal{M}_1$和$\mathcal{M}_2$最大交集的尺寸，$W$是最大权重。该结果改进了Lee、Sidford和Wong [FOCS'15]提出的先前最优$\tilde{O}(nr\log{W})$算法，并成为标准独立或秩预言机模型下首个针对多项式有界权重的次二次算法。本文的主要贡献在于提出了一种高效算法，用于计算加权交换图中的最短路径树。