Faster Algorithms for Mean-Payoff Parity Games

Graph games provide the foundation for modeling and synthesis of reactive processes. Such games are played over graphs where the vertices are controlled by two adversarial players. We consider graph games where the objective of the first player is the conjunction of a qualitative objective (specified as a parity condition) and a quantitative objective (specified as a mean-payoff condition). There are two variants of the problem, namely, the threshold problem where the quantitative goal is to ensure that the mean-payoff value is above a threshold, and the value problem where the quantitative goal is to ensure the optimal mean-payoff value; in both cases ensuring the qualitative parity objective. The previous best-known algorithms for game graphs with $n$ vertices, $m$ edges, parity objectives with $d$ priorities, and maximal absolute reward value $W$ for mean-payoff objectives, are as follows: $O(n^{d+1} \cdot m \cdot W)$ for the threshold problem, and $O(n^{d+2} \cdot m \cdot W)$ for the value problem. Our main contributions are faster algorithms, and the running times of our algorithms are as follows: $O(n^{d-1} \cdot m \cdot W)$ for the threshold problem, and $O(n^{d} \cdot m \cdot W \cdot \log (n\cdot W))$ for the value problem. For mean-payoff parity objectives with two priorities, our algorithms match the best-known bounds of the algorithms for mean-payoff games (without conjunction with parity objectives). Our results are relevant in synthesis of reactive systems with both functional requirement (given as a qualitative objective) and performance requirement (given as a quantitative objective).