Universal Graphs and Good for Games Automata: New Tools for Infinite Duration Games

Abstract

In this paper, we give a self contained presentation of a recent breakthrough in the theory of infinite duration games: the existence of a quasipolynomial time algorithm for solving parity games. We introduce for this purpose two new notions: good for small games automata and universal graphs. The first object, good for small games automata, induces a generic algorithm for solving games by reduction to safety games. We show that it is in a strong sense equivalent to the second object, universal graphs, which is a combinatorial notion easier to reason with. Our equivalence result is very generic in that it holds for all existential memoryless winning conditions, not only for parity conditions.