Solving parity games in big steps

Abstract

Abstract. This paper proposes a new algorithm that improves the complexity bound for solving parity games. Our approach combines McNaughton's iterated fixed point algorithm with a preprocessing step, which is called prior to every recursive call. The preprocessing uses ranking functions similar to Jurdziński's, but with a restricted codomain, to determine all winning regions smaller than a predefined parameter. The combination of the preprocessing step with the recursive call guarantees that McNaughton's algorithm proceeds in big steps, whose size is bounded from below by the chosen parameter. Higher parameters guarantee small call trees, but to the cost of an expensive preprocessing step. An optimal parameter balance the cost of the recursive call and the preprocessing step, resulting in an O(m n γ(c) ) complexity bound for solving parity games with c colors, n positions and m edges, where2 ⌋ if c is even, and γ(c)= c 3 + 1 2 − 1 ⌈ c 2 ⌉⌊ c 2 ⌋ if c is odd.

An Optimal Strategy Improvement Algorithm for Solving Parity and Payoff Games

“…More efficient algorithms for solving parity games will therefore foster the development of performant model checkers and contribute to bringing synthesis techniques to practice. The quest for performant algorithms [1,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] for solving them has therefore been an active field of research during the last decades.…”

“…Traditional forward techniques (≈ O(n 1 2 c ) [16] for parity games with n positions and c colors), backward techniques (≈O(n c ) [12,10,15]), and their combination (≈O(n 1 3 c ) [25]) provide good complexity bounds. However, these bounds are sharp, and techniques with good complexity bounds [25,16] frequently display their worst case complexity on practical examples.…”

“…However, these bounds are sharp, and techniques with good complexity bounds [25,16] frequently display their worst case complexity on practical examples.…”

Distributed Control Synthesis

“…Thus, we can use standard algorithms for parity games with an upper bound on the running time that is exponential only in the number of colors [17,7,27,28]. For controlling the system, we also need a control strategy that enforces the winning condition.…”

“…For a parity objective, it suffices to use a constructive algorithm to solve the explicit parity game, e.g., [27]. We say that a transition t is good in a state s if, and only if, it is in accordance with this strategy.…”