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.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 450, 2016, pp. 43–61.
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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
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DOI: https://doi.org/10.1007/s10958-018-3856-3