Combinatorial simplex algorithms can solve mean payoff games

Xavier Allamigeon 1, 2 Pascal Benchimol 1, 2 Stéphane Gaubert 1, 2 Michael Joswig 3
2 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR
Abstract : A combinatorial simplex algorithm is an instance of the simplex method in which the pivoting depends on combinatorial data only. We show that any algorithm of this kind admits a tropical analogue which can be used to solve mean payoff games. Moreover, any combinatorial simplex algorithm with a strongly polynomial complexity (the existence of such an algorithm is open) would provide in this way a strongly polynomial algorithm solving mean payoff games (all the arithmetic operations being performed on data polynomially bounded in the size of the input, in particular). Mean payoff games are known to be in NP and co-NP; whether they can be solved in polynomial time is an open problem. Our algorithm relies on a tropical implementation of the simplex method over a real closed field of Hahn series. One of the key ingredients is a new scheme for symbolic perturbation which allows us to lift an arbitrary mean payoff game instance into a non-degenerate linear program over Hahn series.
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SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2014, 24 (4), pp.22. 〈10.1137/140953800〉
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https://hal.inria.fr/hal-00930915
Contributeur : Pascal Benchimol <>
Soumis le : mardi 14 janvier 2014 - 16:30:22
Dernière modification le : jeudi 10 mai 2018 - 02:04:13

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Xavier Allamigeon, Pascal Benchimol, Stéphane Gaubert, Michael Joswig. Combinatorial simplex algorithms can solve mean payoff games. SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2014, 24 (4), pp.22. 〈10.1137/140953800〉. 〈hal-00930915〉

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