A cost-scaling algorithm for computing the degree of determinants

Abstract

In this paper, we address computation of the degree deg Det A of Dieudonné determinant Det A ofwhere 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.

A combinatorial algorithm for computing the entire sequence of the maximum degree of minors of a generic partitioned polynomial matrix with $2 \times 2$ submatrices

“…Very recently, Hirai and Ikeda [10] have presented a strongly polynomial-time algorithm for computing deg Det A(t) of A(t) having the following special form…”