In 2021, Casares, Colcombet and Fijalkow introduced the Alternating Cycle Decomposition (ACD), a structure used to define optimal transformations of Muller into parity automata and to obtain theoretical results about the possibility of relabelling automata with different acceptance conditions. In this work, we study the complexity of computing the ACD and its DAG-version, proving that this can be done in polynomial time for suitable representations of the acceptance condition of the Muller automaton. As corollaries, we obtain that we can decide typeness of Muller automata in polynomial time, as well as the parity index of the languages they recognise. Furthermore, we show that we can minimise in polynomial time the number of colours (resp. Rabin pairs) defining a Muller (resp. Rabin) acceptance condition, but that these problems become NP-hard when taking into account the structure of an automaton using such a condition.

翻译：2021年，Casares、Colcombet和Fijalkow引入了交替循环分解（ACD），这是一种用于定义Muller自动机到parity自动机最优转换的结构，并用于获取关于具有不同接受条件的自动机重新标记可能性的理论结果。本文研究了计算ACD及其DAG版本的复杂性，证明对于Muller自动机接受条件的适当表示，这可以在多项式时间内完成。作为推论，我们得出可以在多项式时间内判定Muller自动机的类型性，以及它们所识别语言的parity指数。此外，我们证明可以在多项式时间内最小化定义Muller（或Rabin）接受条件的颜色数（或Rabin对），但当考虑使用此类条件的自动机的结构时，这些问题变为NP难问题。