Favorite Theorems: Parity Games

Cristian Calude, Sanjay Jain, Bakhadyr Khoussainov, Wei Li and Frank Stephan

I covered this theorem in 2017. In a parity game, Alice and Bob take turns walking along a directed graph with integer weights on the vertices and no sinks. Alice wins if the largest weight seen infinitely often is even. While it's not hard to show computing the winner sits in NP\(\cap\)co-NP and even UP\(\cap\)co-UP, the authors give the surprising result that you can determine the winner in near polynomial time.

The result has implications for modal logics, for example that model checking for the \(\mu\)-calculus can now be solved in quasipolynomial-time.

In follow-up work, Hugo Gimbert and Rasmus Ibsen-Jensen give a short proof of correctness of the parity games algorithm. Marcin Jurdziński and Ranko Lazić give an alternative algorithm that reduces the space complexity from quasipolynomial to nearly linear.