A cost-scaling algorithm for computing the degree of determinants

In this paper, we address computation of the degree $\mathop{\rm deg Det} A$ of Dieudonn\'e determinant $\mathop{\rm Det} A$ of $$A = \sum_{k=1}^m A_k x_k t^{c_k},$$ where $A_k$ are $n \times n$ matrices over a field $\mathbb{K}$, $x_k$ are noncommutative variables, $t$ is a variable commuting with $x_k$, $c_k$ 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 $\mathop{\rm deg Det} A$ is obtained by a discrete convex optimization on a Euclidean building. We extend this framework by incorporating a cost scaling technique, and show that $\mathop{\rm deg Det} A$ can be computed in time polynomial of $n,m,\log_2 C$, where $C:= \max_k |c_k|$. We give a polyhedral interpretation of $\mathop{\rm deg Det}$, which says that $\mathop{\rm deg Det} A$ is given by linear optimization over an integral polytope with respect to objective vector $c = (c_k)$. Based on it, we show that our algorithm becomes a strongly polynomial one. We apply this result to an algebraic combinatorial optimization problem arising from a symbolic matrix having $2 \times 2$-submatrix structure.