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Register a new userA symmetric attractor-decomposition lifting algorithm for parity games
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Added by
Pierre Ohlmann
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Progress-measure lifting algorithms for solving parity games have the best worst-case asymptotic runtime, but are limited by their asymmetric nature, and known from the work of Czerwinski et al. (2018) to be subject to a matching quasi-polynomial lower bound inherited from the combinatorics of universal trees. Parys (2019) has developed an ingenious quasi-polynomial McNaughton- Zielonka-style algorithm, and Lehtinen et al. (2019) have improved its worst-case runtime. Jurdzinski and Morvan (2020) have recently brought forward a generic attractor-based algorithm, formalizing a second class of quasi-polynomial solutions to solving parity games, which have runtime quadratic in the size of universal trees. First, we adapt the framework of iterative lifting algorithms to computing attractor-based strategies. Second, we design a symmetric lifting algorithm in this setting, in which two lifting iterations, one for each player, accelerate each other in a recursive fashion. The symmetric algorithm performs at least as well as progress-measure liftings in the worst-case, whilst bypassing their inherent asymmetric limitation. Thirdly, we argue that the behaviour of the generic attractor-based algorithm of Jurdzinski and Morvan (2020) can be reproduced by a specific deceleration of our symmetric lifting algorithm, in which some of the information collected by the algorithm is repeatedly discarded. This yields a novel interpretation of McNaughton-Zielonka-style algorithms as progress-measure lifting iterations (with deliberate set-backs), further strengthening the ties between all known quasi-polynomial algorithms to date.
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In a mean-payoff parity game, one of the two players aims both to achieve a qualitative parity objective and to minimize a quantitative long-term average of payoffs (aka. mean payoff). The game is zero-sum and hence the aim of the other player is to either foil the parity objective or to maximize the mean payoff. Our main technical result is a pseudo-quasi-polynomial algorithm for solving mean-payoff parity games. All algorithms for the problem that have been developed for over a decade have a pseudo-polynomial and an exponential factors in their running times; in the running time of our algorithm the latter is replaced with a quasi-polynomial one. By the results of Chatterjee and Doyen (2012) and of Schewe, Weinert, and Zimmermann (2018), our main technical result implies that there are pseudo-quasi-polynomial algorithms for solving parity energy games and for solving parity games with weights. Our main conceptual contributions are the definitions of strategy decompositions for both players, and a notion of progress measures for mean-payoff parity games that generalizes both parity and energy progress measures. The former provides normal forms for and succinct representations of winning strategies, and the latter enables the application to mean-payoff parity games of the order-theoretic machinery that underpins a recent quasi-polynomial algorithm for solving parity games.
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2.5 player parity games combine the challenges posed by 2.5 player reachability games and the qualitative analysis of parity games. These two types of problems are best approached with different types of algorithms: strategy improvement algorithms for 2.5 player reachability games and recursive algorithms for the qualitative analysis of parity games. We present a method that - in contrast to existing techniques - tackles both aspects with the best suited approach and works exclusively on the 2.5 player game itself. The resulting technique is powerful enough to handle games with several million states.
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