Beyond #CSP: A Dichotomy for Counting Weighted Eulerian Orientations with ARS

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 M\"obius inversion.