A pseudo-quasi-polynomial algorithm for solving mean-payoff parity games

TL;DR

This paper introduces a pseudo-quasi-polynomial algorithm for mean-payoff parity games, improving computational efficiency and generalizing previous methods for solving related game types.

Contribution

It presents a novel pseudo-quasi-polynomial algorithm and new strategy decompositions and progress measures for mean-payoff parity games, advancing the theoretical framework.

Findings

Algorithm runs in pseudo-quasi-polynomial time

Implications for solving parity energy and weighted parity games

Provides normal forms for winning strategies

Abstract

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.