A cost-scaling algorithm for computing the degree of determinants

Abstract

In this paper, we address computation of the degree $\deg {\rm Det} A$ of Dieudonné determinant ${\rm Det} A$ of

$$\begin{aligned} A = \sum_{k=1}^m A_k x_k t^{c_k}, \end{aligned}$$

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 $\deg {\rm Det} A$ is obtained by a discrete convex optimization on a Euclidean building (Hirai 2019). We extend this framework by incorporating a cost-scaling technique and show that $\deg {\rm Det} A$ can be computed in time polynomial of $n,m,\log_2 C$, where $C:= \max_k |c_k|$. We give a polyhedral interpretation of $\deg {\rm Det}$, which says that $\deg {\rm Det}$ 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 also apply our result to an algebraic combinatorial optimization problem arising from a symbolic matrix having $2 \times 2$-submatrix structure.