Abstract
This paper presents an efficient “combinatorial relaxation” algorithm for computing the entire sequence of the maximum degree of minors in rational function matrices, whereas the previous algorithms find them separately for a specified order k. The efficiency of the algorithm is based on the discrete concavity related to valuated bimatroids.



Similar content being viewed by others
References
Commault, C., Dion, J.M.: Structure at infinity of linear multivariable systems: a geometric approach. IEEE Trans. Autom. Control AC–27, 693–696 (1982)
Dress, A.W.M., Wenzel, W.: Valuated matroids. Adv. Math. 93, 214–250 (1992)
Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19(2), 248–264 (1972)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, Berlin (1991)
Iwata, S.: Computing the maximum degree of minors in matrix pencils via combinatorial relaxation. Algorithmica 36, 331–341 (2003)
Iwata, S., Murota, K., Sakuta, I.: Primal-dual combinatorial relaxation algorithms for the maximum degree of subdeterminants. SIAM J. Sci. Comput. 17, 993–1012 (1996)
Iwata, S., Takamatsu, M.: Computing the maximum degree of minors in mixed polynomial matrices via combinatorial relaxation. Algorithmica 66, 346–368 (2013)
Murota, K.: Computing Puiseux-series solutions to determinantal equations via combinatorial relaxation. SIAM J. Comput. 19, 1132–1161 (1990)
Murota, K.: Combinatorial relaxation algorithm for the maximum degree of subdeterminants: computing Smith-McMillan form at infinity and structural indices in Kronecker form. Appl. Algebra Eng. Commun. Comput. 6, 251–273 (1995)
Murota, K.: Computing the degree of determinants via combinatorial relaxation. SIAM J. Comput. 24, 765–796 (1995)
Murota, K.: Finding optimal minors of valuated bimatroids. Appl. Math. Lett. 8(4), 37–41 (1995)
Murota, K.: Matrices and Matroids for Systems Analysis. Springer, Berlin (2000)
Sato, S.: Combinatorial relaxation algorithm for the entire sequence of the maximum degree of minors in mixed polynomial matrices. JSIAM Lett. 7, 49–52 (2015)
Tomizawa, N.: On some techniques useful for solution of transportation network problems. Networks 1, 173–194 (1971)
Acknowledgments
Thanks are due to Kazuo Murota and Takayasu Matsuo for helpful comments on the manuscript. The author would also like to thank the anonymous referee for essential comments. This work is partly supported by JSPS KAKENHI Grant Numbers 25287030, 26280004.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sato, S. Combinatorial Relaxation Algorithm for the Entire Sequence of the Maximum Degree of Minors. Algorithmica 77, 815–835 (2017). https://doi.org/10.1007/s00453-015-0109-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-015-0109-4