A note on the approximation of mean-payoff games

We consider the problem of designing approximation schemes for the values of mean-payoff games. It was recently shown that (1) mean-payoff with rational weights scaled on [−1,1] admit additive fully-polynomial approximation schemes, and (2) mean-payoff games with positive weights admit relative fully-polynomial approximation schemes. We show that the problem of designing additive/relative approximation schemes for general mean-payoff games (i.e. with no constraint on their edge-weights) is P-time equivalent to determining their exact solution.