Parity Games: Another View on Lehtinen's Algorithm

Recently, five quasi-polynomial-time algorithms solving parity games were proposed. We elaborate on one of the algorithms, by Lehtinen (2018). Czerwi\'nski et al. (2019) observe that four of the algorithms can be expressed as constructions of separating automata (of quasi-polynomial size), that is, automata that accept all plays decisively won by one of the players, and rejecting all plays decisively won by the other player. The separating automata corresponding to three of the algorithms are deterministic, and it is clear that deterministic separating automata can be used to solve parity games. The separating automaton corresponding to the algorithm of Lehtinen is nondeterministic, though. While this particular automaton can be used to solve parity games, this is not true for every nondeterministic separating automaton. As a first (more conceptual) contribution, we specify when a nondeterministic separating automaton can be used to solve parity games. We also repeat the correctness proof of the Lehtinen's algorithm, using separating automata. In this part, we prove that her construction actually leads to a faster algorithm than originally claimed in her paper: its complexity is $n^{O(\log n)}$ rather than $n^{O(\log d \cdot \log n)}$ (where $n$ is the number of nodes, and $d$ the number of priorities of a considered parity game), which is similar to complexities of the other quasi-polynomial-time algorithms.