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Computing the Maximum Degree of Minors in Mixed Polynomial Matrices via Combinatorial Relaxation

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Abstract

Mixed polynomial matrices are polynomial matrices with two kinds of nonzero coefficients: fixed constants that account for conservation laws and independent parameters that represent physical characteristics. The computation of their maximum degrees of minors is known to be reducible to valuated independent assignment problems, which can be solved by polynomial numbers of additions, subtractions, and multiplications of rational functions. However, these arithmetic operations on rational functions are much more expensive than those on constants.

In this paper, we present a new algorithm of combinatorial relaxation type. The algorithm finds a combinatorial estimate of the maximum degree by solving a weighted bipartite matching problem, and checks if the estimate is equal to the true value by solving independent matching problems. The algorithm mainly relies on fast combinatorial algorithms and performs numerical computation only when necessary. In addition, it requires no arithmetic operations on rational functions. As a byproduct, this method yields a new algorithm for solving a linear valuated independent assignment problem.

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References

  1. Bareiss, E.H.: Sylvester’s identity and multistep integer-preserving Gaussian elimination. Math. Comput. 22, 565–578 (1968)

    MathSciNet  MATH  Google Scholar 

  2. Bareiss, E.H.: Computational solutions of matrix problems over an integral domain. IMA J. Appl. Math. 10, 68–104 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cantor, D.G., Kaltofen, E.: On fast multiplication of polynomials over arbitrary algebras. Acta Inform. 28, 693–701 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Demmel, J., Kågström, B.: Accurate solutions of ill-posed problems in control theory. SIAM J. Matrix Anal. Appl. 9, 126–145 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dress, A.W.M., Wenzel, W.: Valuated matroids. Adv. Math. 93, 214–250 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gabow, H.N., Xu, Y.: Efficient theoretic and practical algorithms for linear matroid intersection problems. J. Comput. Syst. Sci. 53, 129–147 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gantmacher, F.R.: The Theory of Matrices. Chelsea, New York (1959)

    MATH  Google Scholar 

  8. Harvey, N.J.A.: Algebraic algorithms for matching and matroid problems. SIAM J. Comput. 39, 679–702 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Iwata, S.: Computing the maximum degree of minors in matrix pencils via combinatorial relaxation. Algorithmica 36, 331–341 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Iwata, S., Murota, K.: Combinatorial relaxation algorithm for mixed polynomial matrices. Math. Program. 90, 353–371 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Iwata, S., Takamatsu, M.: On the Kronecker canonical form of mixed matrix pencils. SIAM J. Matrix Anal. Appl. 32, 44–71 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mucha, M., Sankowski, P.: Maximum matchings via Gaussian elimination. In: 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 248–255. IEEE Comput. Soc., Los Alamitos (2004)

    Chapter  Google Scholar 

  14. Mucha, M., Sankowski, P.: Maximum matchings in planar graphs via Gaussian elimination. Algorithmica 45, 3–20 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Murota, K.: Systems Analysis by Graphs and Matroids—Structural Solvability and Controllability. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  16. Murota, K.: Computing Puiseux-series solutions to determinantal equations via combinatorial relaxation. SIAM J. Comput. 19, 1132–1161 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Murota, K.: Computing the degree of determinants via combinatorial relaxation. SIAM J. Comput. 24, 765–796 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Murota, K.: Valuated matroid intersection, I: Optimality criteria. SIAM J. Discrete Math. 9, 545–561 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Murota, K.: Valuated matroid intersection, II: Algorithms. SIAM J. Discrete Math. 9, 562–576 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Murota, K.: On the degree of mixed polynomial matrices. SIAM J. Matrix Anal. Appl. 20, 196–227 (1999)

    Article  MathSciNet  Google Scholar 

  22. Murota, K.: Matrices and Matroids for Systems Analysis. Springer, Berlin (2000)

    MATH  Google Scholar 

  23. Murota, K., Iri, M.: Structural solvability of systems of equations—A mathematical formulation for distinguishing accurate and inaccurate numbers in structural analysis of systems. Jpn. J. Appl. Math. 2, 247–271 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sankowski, P.: Maximum weight bipartite matching in matrix multiplication time. Theor. Comput. Sci. 410, 4480–4488 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Storjohann, A.: Algorithms for matrix canonical forms. Ph.D. thesis, ETH Zürich (2000)

  26. Thorp, J.S.: The singular pencil of a linear dynamical system. Int. J. Control 18, 577–596 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  27. Verghese, G.C., Kailath, T.: Rational matrix structure. IEEE Trans. Autom. Control AC-26, 434–439 (1981)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are grateful to Kazuo Murota for helpful comments on the manuscript. We would also like to thank the anonymous referee for insightful suggestions. This work is supported by a Grant-in-Aid for Scientific Research from the Japan Society for Promotion of Science.

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Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Satoru Iwata

  2. Department of Information and System Engineering, Chuo University, Tokyo, 112-8551, Japan

    Mizuyo Takamatsu

Authors
  1. Satoru Iwata
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  2. Mizuyo Takamatsu
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Corresponding author

Correspondence to Mizuyo Takamatsu.

Additional information

A preliminary version of this paper has appeared in Proceedings of the 15th Conference on Integer Programming and Combinatorial Optimization, LNCS 6655, Springer, 2011, pp. 274–286.

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Iwata, S., Takamatsu, M. Computing the Maximum Degree of Minors in Mixed Polynomial Matrices via Combinatorial Relaxation. Algorithmica 66, 346–368 (2013). https://doi.org/10.1007/s00453-012-9640-8

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  • DOI: https://doi.org/10.1007/s00453-012-9640-8

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