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，卡萨雷斯、科尔康贝和菲亚尔科夫提出了交替环分解（ACD），这是一种用于定义穆勒自动机到奇偶自动机最优变换的结构，并获得了关于使用不同接受条件重标自动机可能性的理论结果。本文研究了计算ACD及其DAG版本的复杂性，证明了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，我们得以在多项式时间内判定穆勒自动机的类型性以及其所识别语言的奇偶指标。此外，我们还证明，定义穆勒（或拉宾）接受条件的颜色数（或拉宾对）可在多项式时间内最小化，但当考虑使用此类条件的自动机结构时，这些问题将变为NP完全问题。