The complexity classification of the Holant problem has remained unresolved for the past fifteen years. Counting complex-weighted Eulerian orientation problems, denoted as \#EO, is regarded as one of the most significant challenges to the comprehensive complexity classification of the Holant problem. This article presents an $\text{FP}^\text{NP}$ vs. \#P dichotomy for \#EO, demonstrating that \#EO defined by a signature set is either \#P-hard or polynomial-time computable with a specific NP oracle. This result provides a comprehensive complexity classification for \#EO, and potentially leads to a dichotomy for the Holant problem. Furthermore, we derive three additional dichotomies related to the Holant problem from the dichotomy for \#EO.

翻译：Holant 问题的复杂性分类在过去十五年中一直悬而未决。计算复加权的欧拉定向问题，记作 #EO，被认为是实现 Holant 问题全面复杂性分类的最重大挑战之一。本文提出了 #EO 的 $\text{FP}^\text{NP}$ 与 #P 二分性，证明了由一组签名定义的 #EO 问题要么是 #P-难的，要么是在特定 NP 预言机下多项式时间可计算的。这一结果为 #EO 提供了全面的复杂性分类，并可能导向 Holant 问题的二分性。此外，我们从 #EO 的二分性中推导出了三个与 Holant 问题相关的额外二分性。