Solving 3-Color Parity Games in $ O(n^2) $ Time

Abstract: Parity games are an expressive framework to consider realizability questions for omega-regular languages. However, it is open whether they can be solved in polynomial time, making them unamenable for practical usage. To overcome this restriction, we consider 3-color parity games, which can be solved in polynomial time. They still cover an expressive fragment of specifications, as they include the classical B\"uchi and co-B\"uchi winning conditions as well as their union and intersection. This already suffices to express many useful combinations of safety and liveness properties, as for example the family of GR(1). The best known algorithm for 3-color parity games solves a game with n vertices in $ O(n^{2}\sqrt{n}) $ time. We improve on this result by presenting a new algorithm, based on simple attractor constructions, which only needs time $ O(n^2) $. As a result, we match the best known running times for solving (co)-B\"uchi games, showing that 3-color parity games are not harder to solve in general.