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Register a new userStrategy Iteration using Non-Deterministic Strategies for Solving Parity Games
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Michael Luttenberger
Publication date
2012
fields
Informatics Engineering
and research's language is
English
Authors
Michael Luttenberger
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This article extends the idea of solving parity games by strategy iteration to non-deterministic strategies: In a non-deterministic strategy a player restricts himself to some non-empty subset of possible actions at a given node, instead of limiting himself to exactly one action. We show that a strategy-improvement algorithm by by Bjoerklund, Sandberg, and Vorobyov can easily be adapted to the more general setting of non-deterministic strategies. Further, we show that applying the heuristic of all profitable switches leads to choosing a locally optimal successor strategy in the setting of non-deterministic strategies, thereby obtaining an easy proof of an algorithm by Schewe. In contrast to the algorithm by Bjoerklund et al., we present our algorithm directly for parity games which allows us to compare it to the algorithm by Jurdzinski and Voege: We show that the valuations used in both algorithm coincide on parity game arenas in which one player can surrender. Thus, our algorithm can also be seen as a generalization of the one by Jurdzinski and Voege to non-deterministic strategies. Finally, using non-deterministic strategies allows us to show that the number of improvement steps is bound from above by O(1.724^n). For strategy-improvement algorithms, this bound was previously only known to be attainable by using randomization.
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Zielonkas classic recursive algorithm for solving parity games is perhaps the simplest among the many existing parity game algorithms. However, its complexity is exponential, while currently the state-of-the-art algorithms have quasipolynomial complexity. Here, we present a modification of Zielonkas classic algorithm that brings its complexity down to $n^{mathcal{O}left(logleft(1+frac{d}{log n}right)right)}$, for parity games of size $n$ with $d$ priorities, in line with previous quasipolynomial-time solutions.
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We study stochastic games with energy-parity objectives, which combine quantitative rewards with a qualitative $omega$-regular condition: The maximizer aims to avoid running out of energy while simultaneously satisfying a parity condition. We show that the corresponding almost-sure problem, i.e., checking whether there exists a maximizer strategy that achieves the energy-parity objective with probability $1$ when starting at a given energy level $k$, is decidable and in $NP cap coNP$. The same holds for checking if such a $k$ exists and if a given $k$ is minimal.
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