On a weighted linear matroid intersection algorithm by deg-det computation
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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 × 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.
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