Combinatorial Simplex Algorithms Can Solve Mean Payoff Games

Abstract

A combinatorial simplex algorithm is an instance of the simplex method in which the pivoting depends on certain 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. Mean payoff games are known to be in ${NP} \cap {co-NP}$; 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 nondegenerate linear program over Hahn series.