From tropical linear algebra to zero-sum games
Résumé
Recently, some relations have appeared between tropical
algebra, linear programming, Perron-Frobenius theory, and zero-sum
games. This talk is devoted to these relations and to their
consequences. In particular, we shall make a connection between two
well known unsolved questions. The first is the existence of a
strongly polynomial pivoting rule in linear programming. Such a rule
would allow us to solve a linear program in a number of arithmetic
operations bounded only by the number of variables and the number of
constraints. The second question is the existence of a polynomial
algorithm to solve mean payoff games. In work with Allamigeon,
Benchimol, and Joswig, we showed that a positive answer to the first
question would yield a positive answer to the second, provided the
pivoting rule satisfies certain conditions. This uses the equivalence
between mean payoff games and tropical linear programs, established in
an earlier work with Akian and Guterman. The proof of these results
will give us the opportunity to see at work non-linear
Perron-Frobenius theory, as well as a number of basic tropical tools:
tropical eigenvalues, extensions or symmetrization of semirings,
tropical analogues of Cramer theorem.