Robust worst cases for parity games algorithms

The McNaughton-Zielonka divide et impera algorithm is the simplest and most flexible approach available in the literature for determining the winner in a parity game. Despite its theoretical exponential worst-case complexity and the negative reputation as a poorly effective algorithm in practice, it has been shown to rank among the best techniques for solving such games. Also, it proved to be resistant to a lower bound attack, even more than the strategy improvements approaches, but finally Friedmann provided a family of games on which the algorithm requires exponential time. An easy analysis of this family shows that a simple memoization technique can help the algorithm solve the family in polynomial time. The same result can also be achieved by exploiting an approach based on the dominion-decomposition techniques proposed in the literature. These observations raise the question whether a suitable combination of dynamic programming and game-decomposition techniques can improve on the exponential worst case of the original algorithm. In this paper we answer this question negatively, by providing a robustly exponential worst case, showing that no possible intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches. The resulting worst case is even more robust than that, since it serves as a lower bound for progress measures based algorithms as well, such as Small Progress Measure and its quasi-polynomial variant recently proposed by Jurdziński and Lazic.