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$n$元素整数权重基集$V$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公共独立集$S^{*} \in \mathcal{I}_1 \cap \mathcal{I}_2$以最大化$S^{*}$的权重。本文提出一个简单的确定性算法，该算法通过$\tilde{O}(nr^{3/4}\log{W})$次秩查询求解加权拟阵交问题，其中$r$为$\mathcal{M}_1$与$\mathcal{M}_2$最大交集的大小，$W$为最大权重。该结果改进了Lee、Sidford和Wong [FOCS'15]先前最优的$\tilde{O}(nr\log{W})$算法，并首次在标准独立性或秩预言机模型下实现了多项式有界权重的次二次算法。本文的主要贡献在于提出一种在加权交换图中高效计算最短路径树的算法。