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-complete when taking into account the structure of an automaton using such a condition.

翻译：2021年，Casares、Colcombet与Fijalkow提出了交替循环分解（ACD），该结构被用于定义穆勒自动机到奇偶自动机的最优转换，并获得了关于在不同接受条件下重新标记自动机可能性的理论结果。在本研究中，我们探讨了计算ACD及其有向无环图版本的复杂度，证明对于穆勒自动机接受条件的适当表示，该计算可在多项式时间内完成。作为推论，我们得出可在多项式时间内判定穆勒自动机的类型性，以及确定其所识别语言的奇偶指数。此外，我们证明了可在多项式时间内最小化定义穆勒（对应地，拉宾）接受条件的颜色数（对应地，拉宾对），但当考虑使用此类接受条件的自动机结构时，这些问题将变为NP完全问题。