A cost-scaling algorithm for computing the degree of determinants
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In this paper, we address computation of the degree deg Det A of Dieudonné determinant Det A of A = ∑_k=1^m A_k x_k t^c_k, where A_k are n × n matrices over a field 𝕂, x_k are noncommutative variables, t is a variable commuting x_k, c_k are integers, and the degree is considered for t. This problem generalizes noncommutative Edmonds' problem and fundamental combinatorial optimization problems including the weighted linear matroid intersection problem. It was shown that deg Det A is obtained by a discrete convex optimization on a Euclidean building. We extend this framework by incorporating a cost scaling technique, and show that deg Det A can be computed in time polynomial of n,m,log_2 C, where C:= max_k |c_k|. We apply this result to an algebraic combinatorial optimization problem arising from a symbolic matrix having 2 × 2-submatrix structure.
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