We define and explore a notion of unique prime factorization for constraint functions, and use this as a new tool to prove a complexity classification for counting weighted Eulerian orientation problems with arrow reversal symmetry (ars). We prove that all such problems are either polynomial-time computable or #P-hard. We show that the class of weighted Eulerian orientation problems subsumes all weighted counting constraint satisfaction problems (#CSP) on Boolean variables. More significantly, we establish a novel connection between #CSP and counting weighted Eulerian orientation problems that is global in nature. This connection is based on a structural determination of all half-weighted affine linear subspaces over ${\mathbb{Z}}_2$, which is proved using Möbius inversion.

我们定义和探索约束函数的唯一素数分解的概念，并将其用作证明具有箭头反转对称性（ars）的加权欧拉定向问题的复杂性分类的新工具。我们证明所有这些问题都是多项式时间可计算的或＃P-hard的。我们表明，加权欧拉定向问题的类别包含布尔变量上的所有加权计数约束满足问题（#CSP）。更重要的是，我们在#CSP和计算加权的欧拉定向问题之间建立了一种新颖的联系，这种联系本质上是全球性的。此连接基于结构上所有半加权仿射线性子空间的结构确定${\mathbb{ž}}_2$，这已通过Möbius反演得到证明。