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