Geodesic
Upper and lower bounds for Grigoriev's algorithm for solving integral tropical linear systems
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part IV, Tome 402 (2012), pp. 69-82

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We investigate an algorithm for solving integral tropical linear systems proposed by D. Yu. Grigoriev in 2010, We give the first nonpolynominal lower bound on time complexity of the algorithm, and also improve known upper bound. 
@article{ZNSL_2012_402_a4,
     author = {A. P. Davydow},
     title = {Upper and lower bounds for {Grigoriev's} algorithm for solving integral tropical linear systems},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {69--82},
     publisher = {mathdoc},
     volume = {402},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_402_a4/}
}
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A. P. Davydow. Upper and lower bounds for Grigoriev's algorithm for solving integral tropical linear systems. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part IV, Tome 402 (2012), pp. 69-82. http://geodesic.mathdoc.fr/item/ZNSL_2012_402_a4/