In this paper, we address the weighted linear matroid intersection problem from the computation of the degree of the determinants of a symbolic matrix. We show that a generic algorithm computing the degree of noncommutative determinants, proposed by the second author, becomes an $O(mn^3 \log n)$ time algorithm for the weighted linear matroid intersection problem, where two matroids are given by column vectors $n \times m$ matrices $A,B$. We reveal that our algorithm is viewed as a "nonstandard" implementation of Frank's weight splitting algorithm for linear matroids. This gives a linear algebraic reasoning to Frank's algorithm. Although our algorithm is slower than existing algorithms in the worst case estimate, it has a notable feature: Contrary to existing algorithms, our algorithm works on different matroids represented by another "sparse" matrices $A^0,B^0$, which skips unnecessary Gaussian eliminations for constructing residual graphs.

中文翻译：

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基于 Deg-Det 计算的加权线性拟阵相交算法

在本文中，我们通过计算符号矩阵的行列式的次数来解决加权线性拟阵相交问题。我们表明，第二作者提出的计算非交换行列式度数的通用算法成为加权线性拟阵相交问题的 $O(mn^3 \log n)$ 时间算法，其中两个拟阵由列向量给出$n \times m$ 个矩阵 $A,B$。我们揭示了我们的算法被视为 Frank 的线性拟阵权重分割算法的“非标准”实现。这为 Frank 算法提供了线性代数推理。尽管我们的算法在最坏情况估计中比现有算法慢，但它有一个显着特点：与现有算法相反，