Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games

TL;DR

This paper demonstrates that combining dynamic programming and game-decomposition techniques cannot prevent exponential worst-case complexity in divide-and-conquer algorithms for parity games, confirming their inherent limitations.

Contribution

It provides a robust exponential worst-case scenario for the McNaughton-Zielonka algorithm, showing that certain enhancements cannot overcome its exponential complexity.

Findings

Simple memoization can solve Friedmann's exponential family in polynomial time.

Decomposition techniques alone do not mitigate the exponential worst case.

No combination of dynamic programming and decomposition techniques can improve the exponential complexity.

Abstract

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 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 the solution of such games. Also, it proved to be resistant to a lower bound attack, even more than the strategy improvements approaches, and only recently a family of games on which the algorithm requires exponential time has been provided by Friedmann. 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 intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches.