  The notion of separating automata was introduced by Bojanczyk and Czerwinski
for understanding the first quasipolynomial time algorithm for parity games. In
this paper we show that separating automata is a powerful tool for constructing
algorithms solving games with combinations of objectives. We construct two new
algorithms: the first for disjunctions of parity and mean payoff objectives,
matching the best known complexity, and the second for disjunctions of mean
payoff objectives, improving on the state of the art. In both cases the
algorithms are obtained through the construction of small separating automata,
using as black boxes the existing constructions for parity objectives and for
mean payoff objectives.
