An Exponential Lower Bound for the Parity Game Strategy Improvement Algorithm as We Know it

This paper presents a new lower bound for the discrete strategy improvement algorithm for solving parity games due to Voege and Jurdzinski. First, we informally show which structures are difficult to solve for the algorithm. Second, we outline a family of games on which the algorithm requires exponentially many strategy iterations, answering in the negative the long-standing question whether this algorithm runs in polynomial time. Additionally we note that the same family of games can be used to prove a similar result w.r.t. the strategy improvement variant by Schewe as well as the strategy iteration for solving discounted payoff games due to Puri.