Deciding Parity Games in asipolynomial Time∗

It is shown that the parity game can be solved in quasipolynomial time. The parameterised parity game – with n nodes andm distinct values (aka colours or priorities) – is proven to be in the class of xed parameter tractable (FPT) problems when parameterised over m. Both results improve known bounds, from runtime nO ( √ n) to O (nlog(m)+6) and from anXP-algorithm with runtimeO (nΘ(m) ) for xed parameterm to an FPT-algorithm with runtimeO (n5) +д(m), for some function д depending on m only. As an application it is proven that coloured Muller games with n nodes andm colours can be decided in time O ((mm · n)5); it is also shown that this bound cannot be improved to O ((2m · n)c ), for any c , unless FPT =W[1].