## Abstract

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

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.

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## Acknowledgements

We thank Kazuo Murota and the referees for comments. We are particularly grateful to one of the referees, who suggested a simpler proof of Proposition 3.2. The first author was supported by JSPS KAKENHI Grant Numbers JP17K00029 and JST PRESTO Grant Number JPMJPR192A, Japan.

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Hirai, H., Ikeda, M. A cost-scaling algorithm for computing the degree of determinants.
*comput. complex.* **31**, 10 (2022). https://doi.org/10.1007/s00037-022-00227-4

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DOI: https://doi.org/10.1007/s00037-022-00227-4