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Complexity of Solving Tropical Linear Systems

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Abstract

An algorithm is designed for solving a tropical linear system with complexity polynomial in the size of the system.

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References

  1. M. Akian, S. Gaubert & A. Guterman (2010). The correspondence between tropical convexity and mean payoff games. Proc. 19 Intern. Symp. Math. Theory of Networks and Systems, Budapest, 1295–1302.

  2. M. Akian, S. Gaubert & A. Guterman (2012). Tropical polyhedra are equivalent to mean payoff games. Internat. J. Algebra Comput. 22(1), 1250001, 43 pp.

    Google Scholar 

  3. Bezem M., Nieuwenhuis R., Rodriguez Carbonell E. (2008) Exponential behaviour of the Butkovic-Zimmermann algorithm for solving two-sided linear systems in max-algebra. Discrete Appl. Math. 156: 3506–3509

    Article  MathSciNet  MATH  Google Scholar 

  4. Bogart T., Jensen A.N., Speyer D., Sturmfels B., Thomas R.R. (2007) Computing tropical varieties. J. Symb. Comput. 42: 54–73

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Butkovic (2010). Max-linear systems: theory and algorithms. Springer.

  6. Butkovic P., Hevery F. (1985) A condition for the strong regularity of matrices in the minimax algebra. Discr. Appl. Math. 11: 209–222

    Article  MathSciNet  MATH  Google Scholar 

  7. Butkovic P., Zimmermann K. (2006) A strongly polynomial algorithm for solving two-sided linear systems in max-algebra. Discrete Appl. Math. 154: 437–446

    Article  MathSciNet  MATH  Google Scholar 

  8. A.P. Davydow (2012). Upper and lower bounds for Grigoriev’s algorithm for solving integral tropical linear systems, Combinatorics and graph theory. Part IV, RuFiDiM’11, Zap. Nauchn. Sem. POMI, POMI, St. Petersburg, 402, 69–82.

  9. Develin M., Santos F., Sturmfels B. (2005) On the rank of a tropical matrix. In Combinatorial and computational geometry, Math. Sci. Res. Inst. Publ., Cambridge Univ. Press 52: 213–242

    MathSciNet  Google Scholar 

  10. Grigoriev D. (1981) Multiplicative complexity of a bilinear form over a commutative ring. Lect. Notes Comput. Sci. 118: 281–286

    Article  Google Scholar 

  11. D. Grigoriev, V.V. Podolskii (2012). Complexity of tropical and min-plus linear prevarieties. arXiv:math/1204.4578.

  12. Izhakian Z., Rowen L. (2009) The tropical rank of a tropical matrix. Communic. Algebra 37: 3912–3927

    Article  MathSciNet  MATH  Google Scholar 

  13. K. H. Kim & F. W. Roush (2005). Factorization of polynomials in one variable over the tropical semiring. arXiv:math/050116/v2.

  14. Kim K.H., Roush F.W. (2006) Kapranov rank vs. tropical rank. Proc. Amer. Math. Soc. 134: 2487–2494

    Article  MathSciNet  MATH  Google Scholar 

  15. Theobald T. (2006) On the frontiers of polynomial computations in tropical geometry. J. Symbolic Comput. 41: 1360–1375

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Dima Grigoriev.

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Grigoriev, D. Complexity of Solving Tropical Linear Systems. comput. complex. 22, 71–88 (2013). https://doi.org/10.1007/s00037-012-0053-5

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  • DOI: https://doi.org/10.1007/s00037-012-0053-5

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