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

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 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 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 possible intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches.

A Parity Game Tale of Two Counters

“…These lower bounds are resistant against techniques like inflation, compression and SCC decomposition, even when applied to each recursive call. Observing that the lower bound of Friedmann can be defeated using memoization, since the number of distinct solved subgames is polynomial, Benerecetti et al proposed a lower bound [5] that is resilient against memoization as well as the other techniques. Their priority promotion algorithm [4] solves these three lower bounds in polynomial time, although this requires inflation for Gazda's lower bound.…”

“…Their priority promotion algorithm [4] solves these three lower bounds in polynomial time, although this requires inflation for Gazda's lower bound. For the original algorithm, two exponential lower bounds are known [5,6]. For the variations of priority promotion, in particular for the delayed promotion policy [5], no lower bound has been published in the literature.…”

“…For the original algorithm, two exponential lower bounds are known [5,6]. For the variations of priority promotion, in particular for the delayed promotion policy [5], no lower bound has been published in the literature. Tangle learning [8] solves all these lower bound examples in polynomial time.…”

Register Games

“…Indeed, if Eve follows such a strategy, then eventually all outputs in the 0register games are 0. Among well-known parity games with register-index 0 are the example games for which strategy improvement and divide-and-conquer algorithms exhibit worst-case complexity [BDM17,Fri09].…”

Attracting Tangles to Solve Parity Games

“…When vertices from a higher α-region are attracted to tangles of player α, progress for player α is reset. Zielonka's algorithm also has no mechanism to store tangles and games that are exponential for Zielonka's algorithm, such as in [4], are trivially solved by tangle learning.…”