Skip to main content
SpringerLink
Log in

An Algorithm for Solving an Overdetermined Tropical Linear System Using the Analysis of Stable Solutions of Subsystems

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

In this paper, we show that an overdetermined tropical linear system has a solution if and only if it contains a square subsystem having a stable solution that is a solution of the original system. This leads to a simple algorithm for solving tropical linear systems in time \( O\left(\left({}_n^m\right)\right){n}^4 \), where m is the number of equations and n is the number of variables. For weakly overdetermined systems, this time is polynomial.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Akian, S. Gaubert, and A. Guterman, “Tropical polyhedra are equivalent to mean payoff games,” Internat. J. Algebra Comput., 22, No. 1 (2012).

  2. P. Butcovič, “Strong regularity of matrices – a survey of results,” Discrete Appl. Math., 48, 45–68 (1994).

    Article  MathSciNet  Google Scholar 

  3. A. Davydow, “New algorithms for solving tropical linear systems,” arXiv:1309.5206 (2013).

  4. D. Grigoriev, “Complexity of solving tropical linear systems,” Comput. Complexity, 22, 71–78 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  5. Z. Izhakian and L. Rowen, “The tropical rank of a tropical matrix,” Comm. Algebra, 37, No. 11, 3912–3927 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  6. H. W. Kuhn, “The Hungarian method for the assignment problem,” Naval Res. Logist., 2, 83–97 (1995).

    Article  MathSciNet  Google Scholar 

  7. J. Richter-Gebert, B. Sturmfels, and T. Theobald, “First steps in tropical geometry,” in: G. L. Litvinov and V. P. Maslov (eds.), Idempotent Mathematics and Mathematical Physics, Amer. Math. Soc., Providence, Rhode Island (2005), pp. 289–317.

  8. S. Reinhard and T. Thorsten, “Combinatorics and genus of tropical intersections and Ehrhart theory,” SIAM J. Discrete Math., 24, No. 1, 17–32 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Felipe Tabera et. al., “Tropical resultants for curves and stable intersection,” Rev. Mat. Iberoamericana, 24, No. 3, 941–961 (2008).

  10. T. Thorsten, “On the frontiers of polynomial computations in tropical geometry,” J. Symbolic Comput., 41, No. 12, 1360–1375 (2006).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg Academic University, St. Petersburg, Russia

    A. Davydow

Authors
  1. A. Davydow
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Davydow.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 450, 2016, pp. 43–61.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davydow, A. An Algorithm for Solving an Overdetermined Tropical Linear System Using the Analysis of Stable Solutions of Subsystems. J Math Sci 232, 25–35 (2018). https://doi.org/10.1007/s10958-018-3856-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-018-3856-3

Search

Navigation